Introduction
Brill–Noether theory studies the linear series on algebraic curves. For a general smooth curve C C C g g g
W d r ( C ) = { L ∈ Pic ( C ) ∣ h 0 ( C , L ) ≥ r + 1 } W^{r}_{d}(C)=\{L\in\operatorname{Pic}(C)\mid h^{0}(C,L)\ge r+1\} W d r ( C ) = { L ∈ Pic ( C ) ∣ h 0 ( C , L ) ≥ r + 1 } in terms of the Brill–Noether number ρ ( g , r , d ) = g − ( r + 1 ) ( g − d + r ) \rho(g,r,d)=g-(r+1)(g-d+r) ρ ( g , r , d ) = g − ( r + 1 ) ( g − d + r ) ρ ( g , r , d ) < 0 \rho(g,r,d)<0 ρ ( g , r , d ) < 0 W d r ( C ) W^{r}_{d}(C) W d r ( C ) ρ ( g , r , d ) ≥ 0 \rho(g,r,d)\ge0 ρ ( g , r , d ) ≥ 0 W d r ( C ) W^{r}_{d}(C) W d r ( C ) ρ ( g , r , d ) \rho(g,r,d) ρ ( g , r , d ) ρ ( g , r , d ) > 0 \rho(g,r,d)>0 ρ ( g , r , d ) > 0 W d r ( C ) W^{r}_{d}(C) W d r ( C ) W d r + 1 ( C ) W^{r+1}_{d}(C) W d r + 1 ( C ) [9 , 10 , 8 ] . A refined Brill–Noether theory seeks an analogous description for curves that are special in moduli, in particular for curves with unexpected maps to projective space, for example for curves with a low degree map to P 1 \mathbb{P}^{1} P 1 k k k π : C → P 1 \pi:C\to\mathbb{P}^{1} π : C → P 1
π ∗ L = O P 1 ( e ⃗ ) : = O P 1 ( e 0 ) ⊕ ⋯ ⊕ O P 1 ( e k − 1 ) \pi_{*}L=\mathcal{O}_{\mathbb{P}^{1}}(\vec{e}):=\mathcal{O}_{\mathbb{P}^{1}}(e_{0})\oplus\cdots\oplus\mathcal{O}_{\mathbb{P}^{1}}(e_{k-1}) π ∗ L = O P 1 ( e ) := O P 1 ( e 0 ) ⊕ ⋯ ⊕ O P 1 ( e k − 1 ) gives the Hurwitz–Brill–Noether loci W e ⃗ ( C ) W^{\vec{e}}(C) W e ( C ) [14 , 15 , 4 , 5 ] . When π : C → P 1 \pi:C\to\mathbb{P}^{1} π : C → P 1 p p p p p p [19 ] , Pflueger considered the refinement given by prescribing the order of vanishing at two points, p , q ∈ C p,q\in C p , q ∈ C [2 ] where analogues of classical Brill–Noether results are proven for transmission loci. In this paper, we give an inductive treatment for the case of one point of ramification, giving new proofs of the results in [2 ] in the case of pointed Hurwitz–Brill–Noether loci W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p )
Let H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^{1},p] [ π : C → P 1 , p ] k k k C C C g g g π \pi π p p p e ⃗ = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k \vec{e}=(e_{0},\ldots,e_{k-1})\in\mathbb{Z}^{k}_{\ge0} e = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) { 0 , k , … , k e 0 , 1 + e 1 , … , 1 + k e 1 , … , ( k − 1 ) + ( k − 1 ) e k − 1 } \{0,k,\ldots,ke_{0},1+e_{1},\ldots,1+ke_{1},\ldots,(k-1)+(k-1)e_{k-1}\} { 0 , k , … , k e 0 , 1 + e 1 , … , 1 + k e 1 , … , ( k − 1 ) + ( k − 1 ) e k − 1 } W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) d = g + ∣ e ⃗ ∣ + k − 1 d=g+|\vec{e}|+k-1 d = g + ∣ e ∣ + k − 1 r = ∣ e ⃗ ∣ + k − 1 r=|\vec{e}|+k-1 r = ∣ e ∣ + k − 1 p p p a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) e ⃗ \vec{e} e a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e )
π ∗ L , π ∗ L ( − p ) , … , π ∗ L ( − ( k − 1 ) p ) . \pi_{*}L,\ \pi_{*}L(-p),\ \ldots,\ \pi_{*}L(-(k-1)p). π ∗ L , π ∗ L ( − p ) , … , π ∗ L ( − ( k − 1 ) p ) . Our first result recovers the one-point case of [2 ] showing that W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p )
Theorem 1 (Theorem 2.5). Let [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^{1},p] [ π : C → P 1 , p ] be a general element of H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot and let e ⃗ ∈ Z ≥ 0 k \vec{e}\in\mathbb{Z}^{k}_{\geq0} e ∈ Z ≥ 0 k Then W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) is nonempty if and only if
g − u ( e ⃗ ) − inv ( e ⃗ ) ≥ 0. g-u(\vec{e})-\operatorname{inv}(\vec{e})\geq0. g − u ( e ) − inv ( e ) ≥ 0. When nonempty, it has dimension g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec{e})-\operatorname{inv}(\vec{e}) g − u ( e ) − inv ( e ) Where u ( e ⃗ ) = ∑ i ≠ j max { 0 , e i − e j − 1 } u(\vec{e})=\sum_{i\ne j}\max\{0,e_i-e_j-1\} u ( e ) = ∑ i = j max { 0 , e i − e j − 1 } and inv e ⃗ = # { ( i , j ) ∣ 0 ≤ i < j ≤ k − 1 , e i < e j } \operatorname{inv}\vec{e}=\#\{(i,j)\mid0\leq i<j\leq k-1,e_i<e_j\} inv e = # {( i , j ) ∣ 0 ≤ i < j ≤ k − 1 , e i < e j }
We note that the assumption that the entries of e ⃗ \vec{e} e π ∗ ( O P 1 ( 1 ) ) = O C ( k p ) \pi^*(\mathcal{O}_{\mathbb{P}^{1}}(1))=\mathcal{O}_{C}(kp) π ∗ ( O P 1 ( 1 )) = O C ( k p ) k p kp k p e ⃗ \vec{e} e inv ( e ⃗ ) = 0 \operatorname{inv}(\vec{e})=0 inv ( e ) = 0 W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) W e ⃗ ( C ) W^{\vec{e}}(C) W e ( C ) [15 ] .
We provide two proofs of this result. For the first proof, we consider the locus of ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) g − u ( e ⃗ ) g-u(\vec{e}) g − u ( e ) [15 ] . Then we see that the additional inversions measure the extra codimension imposed by the marked point via flag degeneracy conditions. This proof has the advantage of being elegant, but relies on the newly introduced concept of nested linear series. The second proof relies on classical flag degeneracy arguments, and on restrictions to open subloci where the ramification orders are exactly as predicted. This proof has the disadvantage of being numerically less elegant, but the advantage of using nothing more than results from [6 ] .
Inductive approach and Brill–Noether analogues
Already due to [2 ] , there is a pointed Hurwitz–Brill–Noether refinement of the results in [15 ] . For a general element [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^{1},p] [ π : C → P 1 , p ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p )
nonempty of pure dimension g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec{e})-\operatorname{inv}(\vec{e}) g − u ( e ) − inv ( e ) g − u ( e ⃗ ) − inv ( e ⃗ ) ≥ 0 g-u(\vec{e})-\operatorname{inv}(\vec{e})\geq0 g − u ( e ) − inv ( e ) ≥ 0 [2 ] and Theorem 2.5 ;
normal and Cohen–Macaulay, and is smooth away from a locus of codimension at least two, see [2 ] and Theorem 3.12 ;
irreducible when g − u ( e ⃗ ) − inv ( e ⃗ ) ≥ 1 g-u(\vec{e})-\operatorname{inv}(\vec{e})\geq1 g − u ( e ) − inv ( e ) ≥ 1 [2 ] ;
finite and reduced when g − u ( e ⃗ ) − inv ( e ⃗ ) = 0 g-u(\vec{e})-\operatorname{inv}(\vec{e})=0 g − u ( e ) − inv ( e ) = 0 [2 ] and Theorem 4.1 .
Aggarwal’s result follow via degeneration to a chain of elliptic curves, while the current paper only consider the degeneration to a curve of genus g − 1 g-1 g − 1
Euler characteristics
The scheme-theoretic description of the elliptic-tail degeneration gives a recursion for the algebraic Euler characteristic. For a general cover of genus g = n + u ( e ⃗ ) + inv ( e ⃗ ) g=n+u(\vec{e})+\operatorname{inv}(\vec{e}) g = n + u ( e ) + inv ( e )
χ n ( e ⃗ ) = χ ( W e ⃗ ( π , p ) , O W e ⃗ ( π , p ) ) . \chi_n(\vec{e})=\chi\left(W^{\vec{e}}(\pi,p),\mathcal{O}_{W^{\vec{e}}(\pi,p)}\right). χ n ( e ) = χ ( W e ( π , p ) , O W e ( π , p ) ) . In particular, for our one-pointed Hurwitz–Brill–Noether loci W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) t i t_i t i e ⃗ \vec{e} e Definition 1.11 , and I ( e ⃗ ) = { i ∣ t i ( e ⃗ ) ≠ e ⃗ } I(\vec{e})=\{i\mid t_i(\vec{e})\ne\vec{e}\} I ( e ) = { i ∣ t i ( e ) = e }
Theorem 2 (Theorem 4.1). The algebraic Euler characteristics satisfy the recursion
χ n ( e ⃗ ) = ∑ i ∈ I ( e ⃗ ) χ n ( t i ( e ⃗ ) ) − I ( e ⃗ ) ⋅ χ n − 1 ( e ⃗ ) . \chi_n(\vec{e})=\sum_{i\in I(\vec{e})}\chi_n(t_i(\vec{e}))-I(\vec{e})\cdot\chi_{n-1}(\vec{e}). χ n ( e ) = i ∈ I ( e ) ∑ χ n ( t i ( e )) − I ( e ) ⋅ χ n − 1 ( e ) . In dimension zero, this specializes to a recursion for cardinalities. Aggarwal already computes these cardinalities more generally by counting reduced words of affine permutations [2 Theorem 1.1(2)] ; our recursion gives an alternative computation for the one-point loci. In positive dimension, it computes the algebraic Euler characteristic, and hence the arithmetic genus when the locus is a curve. This addresses an enumerative direction also raised in [Pf26]. In particular, this formula provides a very efficient algorithm for computing algebraic Euler characteristics (and implicitly genera when g − u ( e ⃗ ) − inv ( e ⃗ ) = 1 g-u(\vec e)-\operatorname{inv}(\vec e)=1 g − u ( e ) − inv ( e ) = 1 [19 ] . As an example, for
e ⃗ = ( 28 , 28 , 18 , 18 , 7 , 2 ) and g = 181 \vec e=(28,28,18,18,7,2) \text{ and } g=181 e = ( 28 , 28 , 18 , 18 , 7 , 2 ) and g = 181 we have u ( e ⃗ ) = 180 u(\vec e)=180 u ( e ) = 180 inv ( e ⃗ ) = 0 \operatorname{inv}(\vec e)=0 inv ( e ) = 0
12354645379354833080765059657446062822690008113927811502151177705426049 ≈ 1.2 × 10 70 . 12354645379354833080765059657446062822690008113927811502151177705426049 \approx1.2 \times10^{70}. 12354645379354833080765059657446062822690008113927811502151177705426049 ≈ 1.2 × 1 0 70 . In § 5 , we give a concrete formula for cardinality and genus for maximal Hurwitz–Brill–Noether loci in gonality 4, and moreover, reduce the problem of computing cardinalities, and more generally algebraic Euler characteristics to an elementary problem of counting duckling walks, and observing duckling relative positions.
Components of expected dimension
Lastly, when k ≥ g + 1 k \ge g+1 k ≥ g + 1 H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot M g , 1 \mathcal{M}_{g,1} M g , 1
M g , d r : = { C ∣ W d r ( C ) ≠ ∅ } \mathcal{M}^{r}_{g,d}:=\{C\mid W^{r}_{d}(C)\ne\varnothing\} M g , d r := { C ∣ W d r ( C ) = ∅ } as Hurwitz–Brill–Noether loci in H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot
1 ≤ − ρ ( g , r , d ) ≤ g − 2 , 1\le-\rho(g,r,d)\le g-2, 1 ≤ − ρ ( g , r , d ) ≤ g − 2 , see [18 ] and [TiB25]. This allow us to prove a similar result for the universal pointed Hurwitz–Brill–Noether spaces
W g e ⃗ : = { [ π : C ⟶ P 1 , p , L ] ∣ [ π , p ] ∈ H g , k t o t , L ∈ W e ⃗ ( π , p ) } . \mathcal{W}^{\vec e}_{g}:=\left\{[\pi:C\longrightarrow\mathbb{P}^{1},p,L]\mid[\pi,p]\in\mathcal{H}^{\mathrm{tot}}_{g,k},\ L\in W^{\vec e}(\pi,p)\right\}. W g e := { [ π : C ⟶ P 1 , p , L ] ∣ [ π , p ] ∈ H g , k tot , L ∈ W e ( π , p ) } . We obtain the following result, which is also Theorem 5.4 .
Theorem 3 (Theorem 5.4). Let g ≥ 1 g\ge1 g ≥ 1 e ⃗ = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k \vec e=(e_{0},\ldots,e_{k-1})\in\mathbb{Z}^{k}_{\ge0} e = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k
1 ≤ c : = u ( e ⃗ ) + inv ( e ⃗ ) − g ≤ k − 3. 1\le c:=u(\vec e)+\operatorname{inv}(\vec e)-g\le k-3. 1 ≤ c := u ( e ) + inv ( e ) − g ≤ k − 3. Write b 0 ( e ⃗ ) < b 1 ( e ⃗ ) < ⋯ < b k − 1 ( e ⃗ ) b_{0}(\vec e)<b_{1}(\vec e)<\cdots<b_{k-1}(\vec e) b 0 ( e ) < b 1 ( e ) < ⋯ < b k − 1 ( e ) k e j + j ke_{j}+j k e j + j 0 ≤ j ≤ k − 1 0\le j\le k-1 0 ≤ j ≤ k − 1 b k − 2 − b c > k b_{k-2}-b_{c}>k b k − 2 − b c > k W g e ⃗ W^{\vec e}_{g} W g e 2 g + k − 3 + ( g − u ( e ⃗ ) − inv ( e ⃗ ) ) 2g+k-3+(g-u(\vec e)-\operatorname{inv}(\vec e)) 2 g + k − 3 + ( g − u ( e ) − inv ( e )) H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot
While technical, the numerical conditions in the theorem are very mild, and only require that e ⃗ \vec e e k e j + j ke_{j}+j k e j + j k k k b k − 2 ( e ⃗ ) b_{k-2}(\vec e) b k − 2 ( e ) e ⃗ \vec e e e 1 > e 2 + 1 e_{1}>e_{2}+1 e 1 > e 2 + 1
Outline
In § 2 , we provide the definition of the pointed Hurwitz–Brill–Noether loci, along with the combinatorial framework. We prove the dimension theorem in § 3 , and establish scheme-theoretic regeneration and discuss the geometric consequences, with the results of [2 ] , in § 4 . In § 5 , we derive the Euler-characteristic recursion and its enumerative applications. Finally, § 6 treats components of expected codimension over the Hurwitz space H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot
Acknowledgments
The authors would like to thank Martin Ulirsch for his insightful comments at the beginning of the project, as well as Daksh Aggarwal, Gavril Farkas, Andres Rojas, and Isabel Vogt for helpful conversations. During part of preparation of this article, A.B. was supported by DFG through the Collaborative Research Centre TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124. A.B. is currently supported by ERC Starting Grant SpiCEl, no.101164820. This research was partially conducted during the time that R.H. was supported by the National Science Foundation under Grant No. DMS-2231565, and under Grant No. DMS-2424556 while the author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Computations on K3 Surfaces and Related Varieties semester program.
AI disclosure
During the preparation of this paper, ChatGPT was used to improve the exposition, assist with bibliography, and referee statements and proofs. The proof and statement of Proposition 5.3 was simplified with assistance from ChatGPT 5.6 and 6, and subsequently simplified and rewritten by the authors. The authors take full responsibility for the mathematical content, references, and final text.
Notation. For notational convenience, if e ⃗ ∈ Z k \vec{e} \in\mathbb{Z}^{k} e ∈ Z k is a vector, we write ∣ e ⃗ ∣ : = ∑ i = 0 k − 1 e i |\vec{e}| := \sum_{i=0}^{k-1} e_i ∣ e ∣ := ∑ i = 0 k − 1 e i
Hurwitz–Brill–Noether space and vanishing orders
In this section we introduce pointed Hurwitz–Brill–Noether loci in terms of the vanishing orders of line bundles at the totally ramified point. We then prove that we have an equivalent definition in terms of the splitting types of
π ∗ L , π ∗ ( L − p ) , … , π ∗ ( L − ( k − 1 ) ⋅ p ) . \pi_*L,\ \pi_*(L-p),\ \ldots,\ \pi_*(L-(k-1)\cdot p). π ∗ L , π ∗ ( L − p ) , … , π ∗ ( L − ( k − 1 ) ⋅ p ) . The goal of this section will then be to provide the combinatorics necessary in order to compute the dimension of pointed Hurwitz–Brill–Noether loci.
Let r ≥ 1 r \ge1 r ≥ 1 d = g + r d=g+r d = g + r
a = ( 0 ≤ a 0 < a 1 < ⋯ < a r ≤ d ) \mathbf{a}=(0\le a_0<a_1<\cdots<a_r\le d) a = ( 0 ≤ a 0 < a 1 < ⋯ < a r ≤ d ) be a sequence of vanishing orders. For a pointed curve [ C , p ] ∈ M g , 1 [C,p]\in\mathcal{M}_{g,1} [ C , p ] ∈ M g , 1
W a , d r ( C , p ) : = { L ∈ Pic d ( C ) ∣ h 0 ( C , L ( − a i ⋅ p ) ) ≥ r + 1 − i for 0 ≤ i ≤ r } . W^r_{\mathbf{a},d}(C,p):=\{L\in\operatorname{Pic}^{d}(C)\mid h^0(C,L(-a_i\cdot p))\ge r+1-i\ \text{for}\ 0\le i\le r\}. W a , d r ( C , p ) := { L ∈ Pic d ( C ) ∣ h 0 ( C , L ( − a i ⋅ p )) ≥ r + 1 − i for 0 ≤ i ≤ r } . In particular, we ask what is the dimension of this space when C C C k k k π : C → P 1 \pi:C\to\mathbb{P}^1 π : C → P 1 p p p
In order to compute this dimension, we need to understand what happens when we degenerate the curve C C C g − 1 g-1 g − 1
Let now [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^1,p] [ π : C → P 1 , p ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot p p p L L L C C C
Definition 1.1. A sequence of vanishing orders a = ( a 0 , a 1 , … , a r ) \mathbf{a}=(a_0,a_1,\ldots,a_r) a = ( a 0 , a 1 , … , a r ) k k k
There is an immediate way to obtain k k k e ⃗ \vec{e} e
Definition 1.2. Let e ⃗ = ( e 0 , e 1 , … , e k − 1 ) ∈ Z ≥ 0 k \vec{e}=(e_0,e_1,\ldots,e_{k-1})\in\mathbb{Z}_{\ge0}^{k} e = ( e 0 , e 1 , … , e k − 1 ) ∈ Z ≥ 0 k k k k a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e )
{ 0 , k , … , e 0 ⋅ k } ∪ { 1 , k + 1 , … , e 1 ⋅ k + 1 } ∪ ⋯ ∪ { k − 1 , 2 k − 1 , … , e k − 1 ⋅ k + k − 1 } \{0,k,\ldots,e_0\cdot k\}\cup\{1,k+1,\ldots,e_1\cdot k+1\}\cup\cdots\cup\{k-1,2k-1,\ldots,e_{k-1}\cdot k+k-1\} { 0 , k , … , e 0 ⋅ k } ∪ { 1 , k + 1 , … , e 1 ⋅ k + 1 } ∪ ⋯ ∪ { k − 1 , 2 k − 1 , … , e k − 1 ⋅ k + k − 1 } in increasing order.
For [ π : C → P 1 , p ] ∈ H g , k t o t [\pi:C\to\mathbb{P}^1,p]\in\mathcal{H}^{\mathrm{tot}}_{g,k} [ π : C → P 1 , p ] ∈ H g , k tot W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) W g + r , a ( e ⃗ ) r , ∘ ( C , p ) W^{r,\circ}_{g+r,\mathbf{a}(\vec{e})}(C,p) W g + r , a ( e ) r , ∘ ( C , p ) g + r g+r g + r C C C p p p a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e )
Example 1.3. We give an example of the vanishing orders for k = 5 k=5 k = 5 e ⃗ = ( 5 , 4 , 0 , 1 , 0 ) \vec{e}=(5,4,0,1,0) e = ( 5 , 4 , 0 , 1 , 0 )
a ( 5 , 4 , 0 , 1 , 0 ) = ( 0 , 1 , 2 , 3 , 4 , 5 , 6 , 8 , 10 , 11 , 15 , 16 , 20 , 21 , 25 ) . \mathbf{a}(5,4,0,1,0)=(0,1,2,3,4,5,6,8,10,11,15,16,20,21,25). a ( 5 , 4 , 0 , 1 , 0 ) = ( 0 , 1 , 2 , 3 , 4 , 5 , 6 , 8 , 10 , 11 , 15 , 16 , 20 , 21 , 25 ) . The vanishing orders in a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) e i + 1 e_i+1 e i + 1 a ( 5 , 4 , 0 , 1 , 0 ) \mathbf{a}(5,4,0,1,0) a ( 5 , 4 , 0 , 1 , 0 ) a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e )
Figure 1. Vanishing orders at p p p L ∈ W ( 5 , 4 , 0 , 1 , 0 ) ( π , p ) L \in W(5,4,0,1,0)(\pi,p) L ∈ W ( 5 , 4 , 0 , 1 , 0 ) ( π , p )
We aim to show that the splitting types of line bundles in the loci W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) e ⃗ \vec{e} e
We need the following result about morphisms of vector bundles on P 1 \mathbb{P}^1 P 1
Lemma 1.5. Let e ⃗ , f ⃗ ∈ Z k \vec{e},\vec{f} \in\mathbb{Z}^k e , f ∈ Z k ∣ e ⃗ ∣ − 1 = ∣ f ⃗ ∣ |\vec{e}|-1=|\vec{f}| ∣ e ∣ − 1 = ∣ f ∣ O ( f ⃗ ) ↪ O ( e ⃗ ) \mathcal{O}(\vec{f}) \hookrightarrow\mathcal{O}(\vec{e}) O ( f ) ↪ O ( e ) P 1 \mathbb{P}^1 P 1 f ⃗ \vec{f} f e ⃗ \vec{e} e 1 1 1
Proof. By twisting the morphism O ( f ⃗ ) → O ( e ⃗ ) \mathcal{O}(\vec{f}) \to\mathcal{O}(\vec{e}) O ( f ) → O ( e ) e ⃗ \vec{e} e f ⃗ \vec{f} f 1 1 1 N N N e ⃗ \vec{e} e
Recall that for n > m n>m n > m Hom ( O P 1 ( n ) , O P 1 ( m ) ) = 0 \operatorname{Hom}(\mathcal{O}_{\mathbb{P}^1}(n),\mathcal{O}_{\mathbb{P}^1}(m))=0 Hom ( O P 1 ( n ) , O P 1 ( m )) = 0 n n n
O ( f ⃗ ≥ n ) → O ( e ⃗ ≥ n ) \mathcal{O}(\vec{f}_{\ge n}) \to\mathcal{O}(\vec{e}_{\ge n}) O ( f ≥ n ) → O ( e ≥ n ) In particular, we have the inequality of cardinalities
# { i ∣ f i ≥ n } ≤ # { i ∣ e i ≥ n } for every n ≥ 1. \#\{i \mid f_i \ge n\} \le\#\{i \mid e_i \ge n\} \text{ for every } n \ge1. # { i ∣ f i ≥ n } ≤ # { i ∣ e i ≥ n } for every n ≥ 1. We denote
b n : = # { i ∣ e i = n } and c n : = # { i ∣ f i = n } . b_n := \#\{i \mid e_i=n\} \quad\text{and}\quad c_n := \#\{i \mid f_i=n\}. b n := # { i ∣ e i = n } and c n := # { i ∣ f i = n } . The condition
∑ i = 0 k − 1 e i − 1 = ∑ i = 0 k − 1 f i \sum_{i=0}^{k-1} e_i-1=\sum_{i=0}^{k-1} f_i i = 0 ∑ k − 1 e i − 1 = i = 0 ∑ k − 1 f i can be rewritten as
∑ n = 1 N n ⋅ b n − 1 = ∑ n = 1 N n ⋅ c n \sum_{n=1}^{N} n\cdot b_n-1=\sum_{n=1}^{N} n\cdot c_n n = 1 ∑ N n ⋅ b n − 1 = n = 1 ∑ N n ⋅ c n or equivalently as
∑ n = 1 N ∑ j = n N b j − 1 = ∑ n = 1 N ∑ j = n N c j . \sum_{n=1}^{N}\sum_{j=n}^{N} b_j - 1 = \sum_{n=1}^{N}\sum_{j=n}^{N} c_j. n = 1 ∑ N j = n ∑ N b j − 1 = n = 1 ∑ N j = n ∑ N c j . But for any 1 ≤ n ≤ N 1 \le n \le N 1 ≤ n ≤ N
∑ j = n N c j ≤ ∑ j = n N b j . \sum_{j=n}^{N} c_j \le\sum_{j=n}^{N} b_j. j = n ∑ N c j ≤ j = n ∑ N b j . In particular, all the inequalities but one are in fact equalities. This implies that f ⃗ \vec{f} f
Using Lemma 1.5 , we can study pointed Brill–Noether conditions on [ π : C → P 1 , p ] ∈ H g , k t o t [\pi: C \to\mathbb{P}^{1},p] \in\mathcal{H}^{\mathrm{tot}}_{g,k} [ π : C → P 1 , p ] ∈ H g , k tot L L L C C C p p p k k k
Proposition 1.6. Let [ π : C → P 1 , p ] [\pi: C \to\mathbb{P}^{1},p] [ π : C → P 1 , p ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot L L L C C C a ≥ k a \ge k a ≥ k H 0 ( C , L ) H^{0}(C,L) H 0 ( C , L ) p p p a − k a-k a − k
Proof. We denote by e ⃗ 0 , e ⃗ 1 , … , e ⃗ k \vec{e}_{0},\vec{e}_{1},\ldots,\vec{e}_{k} e 0 , e 1 , … , e k
π ∗ ( L ( − i p ) ) ≅ O ( e ⃗ i ) for every 0 ≤ i ≤ k . \pi_{*}(L(-ip)) \cong\mathcal{O}(\vec{e}_{i}) \text{ for every } 0 \le i \le k. π ∗ ( L ( − i p )) ≅ O ( e i ) for every 0 ≤ i ≤ k . Notice that e ⃗ i \vec{e}_{i} e i e ⃗ i + 1 \vec{e}_{i+1} e i + 1 e ⃗ i \vec{e}_{i} e i Lemma 1.5 .
We write a = i k + r a=ik+r a = ik + r 0 ≤ r ≤ k − 1 0 \le r \le k-1 0 ≤ r ≤ k − 1 a a a H 0 ( C , L ) H^{0}(C,L) H 0 ( C , L )
h 0 ( C , L ( − a p ) ) = h 0 ( C , L ( − ( a + 1 ) p ) ) + 1 h^{0}(C,L(-ap))=h^{0}(C,L(-(a+1)p))+1 h 0 ( C , L ( − a p )) = h 0 ( C , L ( − ( a + 1 ) p )) + 1 and hence, via pushforward and the push-pull formula,
h 0 ( P 1 , O ( e ⃗ r ) ⊗ O ( − i ) ) = h 0 ( P 1 , O ( e ⃗ r + 1 ) ⊗ O ( − i ) ) + 1. h^{0}(\mathbb{P}^{1},\mathcal{O}(\vec{e}_{r})\otimes\mathcal{O}(-i))=h^{0}(\mathbb{P}^{1},\mathcal{O}(\vec{e}_{r+1})\otimes\mathcal{O}(-i))+1. h 0 ( P 1 , O ( e r ) ⊗ O ( − i )) = h 0 ( P 1 , O ( e r + 1 ) ⊗ O ( − i )) + 1. Because e ⃗ r \vec{e}_{r} e r e ⃗ r + 1 \vec{e}_{r+1} e r + 1 e ⃗ r \vec{e}_{r} e r i i i
Completely analogously, a − k a-k a − k e ⃗ r \vec{e}_{r} e r e ⃗ r + 1 \vec{e}_{r+1} e r + 1 i − 1 i-1 i − 1 i i i i − 1 i-1 i − 1
Example 1.7. Given L ∈ W e ⃗ ( π , p ) L \in W^{\vec{e}}(\pi,p) L ∈ W e ( π , p ) L ( − i p ) L(-ip) L ( − i p ) L ( − p ) L(-p) L ( − p ) a ( e ⃗ ) a(\vec{e}) a ( e ) Figure 2 .
Figure 2. Vanishing orders of L ( − p ) L(-p) L ( − p ) L ∈ W ( 5 , 4 , 0 , 1 , 0 ) ( π , p ) L \in W^{(5,4,0,1,0)}(\pi,p) L ∈ W ( 5 , 4 , 0 , 1 , 0 ) ( π , p )
Thus, for L ∈ W ( 5 , 4 , 0 , 1 , 0 ) ( π , p ) L \in W^{(5,4,0,1,0)}(\pi,p) L ∈ W ( 5 , 4 , 0 , 1 , 0 ) ( π , p ) L ( − p ) L(-p) L ( − p )
( 0 , 1 , 2 , 3 , 4 , 5 , 7 , 9 , 10 , 14 , 15 , 19 , 20 , 24 ) , (0,1,2,3,4,5,7,9,10,14,15,19,20,24), ( 0 , 1 , 2 , 3 , 4 , 5 , 7 , 9 , 10 , 14 , 15 , 19 , 20 , 24 ) , which is just the vanishing sequence corresponding to the vector ( 4 , 0 , 1 , 0 , 4 ) (4,0,1,0,4) ( 4 , 0 , 1 , 0 , 4 )
To keep track of the vanishing orders, we consider the following definition.
Definition 1.8. Given e ⃗ = ( e 0 , … , e k − 1 ) ∈ Z k \vec{e}=(e_0,\ldots,e_{k-1})\in\mathbb{Z}^k e = ( e 0 , … , e k − 1 ) ∈ Z k
σ : Z k → Z k , e ⃗ = ( e 0 , e 1 , e 2 , … , e k − 1 ) ↦ ( e 1 , e 2 , … , e k − 1 , e 0 − 1 ) . \sigma:\mathbb{Z}^k\to\mathbb{Z}^k,\quad\vec{e}=(e_0,e_1,e_2,\ldots,e_{k-1})\mapsto(e_1,e_2,\ldots,e_{k-1},e_0-1). σ : Z k → Z k , e = ( e 0 , e 1 , e 2 , … , e k − 1 ) ↦ ( e 1 , e 2 , … , e k − 1 , e 0 − 1 ) . We define σ i : Z k → Z k \sigma^i:\mathbb{Z}^k\to\mathbb{Z}^k σ i : Z k → Z k i th i^{\text{th}} i th σ \sigma σ
σ 2 ( e ⃗ ) = ( e 2 , e 3 , … , e k − 1 , e 0 − 1 , e 1 − 1 ) , and σ − 1 ( e ⃗ ) = ( e k − 1 + 1 , e 0 , e 1 , … , e k − 2 ) . \sigma^2(\vec{e})=(e_2,e_3,\ldots,e_{k-1},e_0-1,e_1-1),\quad\text{and}\quad\sigma^{-1}(\vec{e})=(e_{k-1}+1,e_0,e_1,\ldots,e_{k-2}). σ 2 ( e ) = ( e 2 , e 3 , … , e k − 1 , e 0 − 1 , e 1 − 1 ) , and σ − 1 ( e ) = ( e k − 1 + 1 , e 0 , e 1 , … , e k − 2 ) . Remark 1.9. Let L ∈ Pic ( C ) L\in\operatorname{Pic}(C) L ∈ Pic ( C ) O C ( k p ) ≅ π ∗ O P 1 ( 1 ) \mathcal{O}_C(kp)\cong\pi^*\mathcal{O}_{\mathbb{P}^1}(1) O C ( k p ) ≅ π ∗ O P 1 ( 1 )
h 0 ( P 1 , π ∗ ( L ( − i p ) ) ⊗ O P 1 ( n ) ) = h 0 ( C , L ( ( n k − i ) p ) ) h^0\left(\mathbb{P}^1,\pi_*(L(-ip))\otimes\mathcal{O}_{\mathbb{P}^1}(n)\right)=h^0(C,L((nk-i)p)) h 0 ( P 1 , π ∗ ( L ( − i p )) ⊗ O P 1 ( n ) ) = h 0 ( C , L (( nk − i ) p )) for all i , n ∈ Z i,n\in\mathbb{Z} i , n ∈ Z
H i ( n ) : = h 0 ( P 1 , π ∗ ( L ( − i p ) ) ⊗ O P 1 ( n ) ) H_i(n):=h^0\left(\mathbb{P}^1,\pi_*(L(-ip))\otimes\mathcal{O}_{\mathbb{P}^1}(n)\right) H i ( n ) := h 0 ( P 1 , π ∗ ( L ( − i p )) ⊗ O P 1 ( n ) ) determines the splitting type of π ∗ ( L ( − i p ) ) \pi_*(L(-ip)) π ∗ ( L ( − i p )) O P 1 ( j ) \mathcal{O}_{\mathbb{P}^1}(j) O P 1 ( j ) π ∗ ( L ( − i p ) ) \pi_*(L(-ip)) π ∗ ( L ( − i p ))
H i ( − j ) − 2 H i ( − j − 1 ) + H i ( − j − 2 ) ; H_i(-j)-2H_i(-j-1)+H_i(-j-2); H i ( − j ) − 2 H i ( − j − 1 ) + H i ( − j − 2 ) ; see [14 Section 2] and [7 Lemma 5.6] .
In particular, the vanishing orders at p p p L ( − i p ) L(-ip) L ( − i p ) i ∈ Z i\in\mathbb{Z} i ∈ Z L ( − i p ) L(-ip) L ( − i p ) i ∈ Z i\in\mathbb{Z} i ∈ Z L L L p p p
Consequently for L ∈ W e ⃗ ( π , p ) L\in W^{\vec{e}}(\pi,p) L ∈ W e ( π , p ) L ( − i p ) L(-ip) L ( − i p ) e ⃗ \vec{e} e p p p
Proposition 1.10. For e ⃗ ∈ Z ≥ 0 k \vec{e}\in\mathbb{Z}^k_{\geq0} e ∈ Z ≥ 0 k
W e ⃗ ( π , p ) = { L ∈ Pic ( C ) | π ∗ ( L ( − i p ) ) ≅ O P 1 ( σ i ( e ⃗ ) ) for every 0 ≤ i ≤ k − 1 } . W^{\vec{e}}(\pi,p)=\left\{L\in\operatorname{Pic}(C)\ \middle|\ \pi_*(L(-ip))\cong\mathcal{O}_{\mathbb{P}^1}\left(\sigma^i(\vec{e})\right)\text{ for every }0\leq i\leq k-1\right\}. W e ( π , p ) = { L ∈ Pic ( C ) π ∗ ( L ( − i p )) ≅ O P 1 ( σ i ( e ) ) for every 0 ≤ i ≤ k − 1 } . Equivalently, the condition on the right-hand side holds for every i ∈ Z i\in\mathbb{Z} i ∈ Z
Proof. The inclusion ⊇ \supseteq ⊇ L ∈ W e ⃗ ( π , p ) L\in W^{\vec{e}}(\pi,p) L ∈ W e ( π , p )
π ∗ ( L ( − p ) ) ≅ O P 1 ( σ ( e ⃗ ) ) . \pi_*(L(-p))\cong\mathcal{O}_{\mathbb{P}^1}(\sigma(\vec{e})). π ∗ ( L ( − p )) ≅ O P 1 ( σ ( e )) . Repeating the argument, we obtain the first inclusion.
Using Remark 1.9, the vanishing orders of L L L
π ∗ L , π ∗ ( L − p ) , … , π ∗ ( L − ( k − 1 ) ⋅ p ) . \pi_*L,\ \pi_*(L-p),\ldots,\pi_*(L-(k-1)\cdot p). π ∗ L , π ∗ ( L − p ) , … , π ∗ ( L − ( k − 1 ) ⋅ p ) . In particular, the inclusion above tells us that the conditions
π ∗ ( L ( − i p ) ) ≅ O P 1 ( σ i ( e ⃗ ) ) for every 0 ≤ i ≤ k − 1 \pi_*(L(-ip))\cong\mathcal{O}_{\mathbb{P}^1}\left(\sigma^i(\vec{e})\right)\text{ for every }0\leq i\leq k-1 π ∗ ( L ( − i p )) ≅ O P 1 ( σ i ( e ) ) for every 0 ≤ i ≤ k − 1 determine the same vanishing orders for L L L L ∈ W e ⃗ ( π , p ) L\in W^{\vec{e}}(\pi,p) L ∈ W e ( π , p ) 0 ≤ i ≤ k − 1 0\leq i\leq k-1 0 ≤ i ≤ k − 1
π ∗ ( L ( n k p ) ) = ( π ∗ L ) ⊗ O P 1 ( n ) \pi_*(L(nkp))=(\pi_*L)\otimes\mathcal{O}_{\mathbb{P}^1}(n) π ∗ ( L ( nk p )) = ( π ∗ L ) ⊗ O P 1 ( n ) for any n n n L L L C C C
We study degenerations of pointed Hurwitz–Brill–Noether loci over the admissible cover compactification H ‾ g , k \overline{\mathcal{H}}_{g,k} H g , k H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot
Setup 1.1 (Elliptic-tail degeneration). Fix an admissible cover
π 0 : X ∪ q E ⟶ P X ∪ q P E , \pi_0: X \cup_q E \longrightarrow P_X \cup_q P_E, π 0 : X ∪ q E ⟶ P X ∪ q P E , where X X X E E E g − 1 g-1 g − 1 1 1 1 P X ≅ P E ≅ P 1 P_X \cong P_E \cong\mathbb{P}^1 P X ≅ P E ≅ P 1
π X : X ⟶ P X , π E : E ⟶ P E \pi_X: X \longrightarrow P_X,\qquad\pi_E: E \longrightarrow P_E π X : X ⟶ P X , π E : E ⟶ P E have degree k k k q q q p ∈ E ∖ { q } p \in E \setminus\{q\} p ∈ E ∖ { q } π E \pi_E π E p − q ∈ Pic 0 ( E ) p-q \in\operatorname{Pic}^{0}(E) p − q ∈ Pic 0 ( E ) k k k
[ π X : X ⟶ P X , q ] ∈ H g − 1 , k t o t [\pi_X: X \longrightarrow P_X,q] \in\mathcal{H}^{\mathrm{tot}}_{g-1,k} [ π X : X ⟶ P X , q ] ∈ H g − 1 , k tot is general.
To describe the possible vanishing orders at the node q q q
Definition 1.11. For any vector e ⃗ = ( e 0 , … , e k − 1 ) \vec{e}=(e_0,\ldots,e_{k-1}) e = ( e 0 , … , e k − 1 ) 0 ≤ i ≤ k − 1 0\leq i\leq k-1 0 ≤ i ≤ k − 1 t j ( e ⃗ ) t_j(\vec{e}) t j ( e )
For 1 ≤ j ≤ k − 1 1\leq j\leq k-1 1 ≤ j ≤ k − 1
t j ( e ⃗ ) : = { ( e 0 , e 1 , … , e j − 2 , e j , e j − 1 , e j + 1 , … , e k − 1 ) , if e j > e j − 1 , e ⃗ , otherwise. t_j(\vec{e}) :=
\begin{cases}
(e_0,e_1,\ldots,e_{j-2},e_j,e_{j-1},e_{j+1},\ldots,e_{k-1}), & \text{if } e_j>e_{j-1},\\
\vec{e}, & \text{otherwise.}
\end{cases} t j ( e ) := { ( e 0 , e 1 , … , e j − 2 , e j , e j − 1 , e j + 1 , … , e k − 1 ) , e , if e j > e j − 1 , otherwise. For j = 0 j=0 j = 0
t 0 ( e ⃗ ) : = { ( e k − 1 + 1 , e 1 , … , e k − 2 , e 0 − 1 ) , if e 0 > e k − 1 + 1 , e ⃗ , otherwise. t_0(\vec{e}) :=
\begin{cases}
(e_{k-1}+1,e_1,\ldots,e_{k-2},e_0-1), & \text{if } e_0>e_{k-1}+1,\\
\vec{e}, & \text{otherwise.}
\end{cases} t 0 ( e ) := { ( e k − 1 + 1 , e 1 , … , e k − 2 , e 0 − 1 ) , e , if e 0 > e k − 1 + 1 , otherwise. Lemma 1.12. Let [ π 0 : X ∪ q E ⟶ P X ∪ q P E , p ] ∈ H g , k t o t [\pi_0:X\cup_q E\longrightarrow P_X\cup_q P_E,p]\in\mathcal{H}^{\mathrm{tot}}_{g,k} [ π 0 : X ∪ q E ⟶ P X ∪ q P E , p ] ∈ H g , k tot e ⃗ ∈ Z ≥ 0 k \vec{e}\in\mathbb{Z}_{\geq0}^k e ∈ Z ≥ 0 k r = k + ∣ e ⃗ ∣ − 1 r=k+|\vec{e}|-1 r = k + ∣ e ∣ − 1 d = g + r d=g+r d = g + r
a ( e ⃗ ) = ( 0 ≤ a 0 < a 1 < ⋯ < a r ≤ d ) a(\vec{e})=(0\leq a_0<a_1<\cdots<a_r\leq d) a ( e ) = ( 0 ≤ a 0 < a 1 < ⋯ < a r ≤ d ) the k k k e ⃗ \vec{e} e { l X : = ( L X , V X ) , l E : = ( L E , V E ) } \{l_X:=(L_X,V_X),l_E:=(L_E,V_E)\} { l X := ( L X , V X ) , l E := ( L E , V E )} g d r g^r_d g d r a ( e ⃗ ) a(\vec{e}) a ( e ) p p p { L X , L E } ∈ Pic d ( X ) × Pic d ( E ) \{L_X,L_E\}\in\operatorname{Pic}^d(X)\times\operatorname{Pic}^d(E) { L X , L E } ∈ Pic d ( X ) × Pic d ( E )
( O X ( q ) ⊗ W r + g − 1 , a ( e ⃗ ) r ( X , q ) ) × Pic d ( E ) \left(\mathcal{O}_X(q)\otimes W^{r}_{r+g-1,a(\vec{e})}(X,q)\right)\times\operatorname{Pic}^d(E) ( O X ( q ) ⊗ W r + g − 1 , a ( e ) r ( X , q ) ) × Pic d ( E ) and
( O X ( q ) ⊗ W r + g − 1 , a ( t j ( e ⃗ ) ) r ( X , q ) ) × { O E ( j p + ( d − j ) q ) } , \left(\mathcal{O}_X(q)\otimes W^{r}_{r+g-1,a(t_j(\vec{e}))}(X,q)\right)\times\{\mathcal{O}_E(jp+(d-j)q)\}, ( O X ( q ) ⊗ W r + g − 1 , a ( t j ( e )) r ( X , q ) ) × { O E ( j p + ( d − j ) q )} , for 0 ≤ j ≤ k − 1 0\leq j\leq k-1 0 ≤ j ≤ k − 1 t j ( e ⃗ ) ≠ e ⃗ t_j(\vec{e})\neq\vec{e} t j ( e ) = e
Proof. We distinguish two possible cases.
Case 1: If L E L_E L E O E ( j p + ( d − j ) q ) \mathcal{O}_E(jp+(d-j)q) O E ( j p + ( d − j ) q ) 0 ≤ j ≤ k − 1 0\leq j\leq k-1 0 ≤ j ≤ k − 1 E E E q q q
( d − 1 − a r , … , d − 1 − a 0 ) . (d-1-a_r,\ldots,d-1-a_0). ( d − 1 − a r , … , d − 1 − a 0 ) . This implies, using the definition of limit linear series, that the vanishing orders of l X l_X l X q q q
1 + a ( e ⃗ ) : = ( a 0 + 1 , … , a r + 1 ) . 1+a(\vec{e}):=(a_0+1,\ldots,a_r+1). 1 + a ( e ) := ( a 0 + 1 , … , a r + 1 ) . It follows that ( L X ( − q ) , V X ( − q ) ) (L_X(-q),V_X(-q)) ( L X ( − q ) , V X ( − q )) g d − 1 r g^{r}_{d-1} g d − 1 r X X X a ( e ⃗ ) a(\vec{e}) a ( e ) q q q
Case 2: If L E L_E L E O E ( j p + ( d − j ) q ) \mathcal{O}_E(jp+(d-j)q) O E ( j p + ( d − j ) q ) 0 ≤ j ≤ k − 1 0\leq j\leq k-1 0 ≤ j ≤ k − 1 E E E q q q
( d − 1 − a r + δ a r , j , … , d − 1 − a 0 + δ a 0 , j ) , (d-1-a_r+\delta^{a_r,j},\ldots,d-1-a_0+\delta^{a_0,j}), ( d − 1 − a r + δ a r , j , … , d − 1 − a 0 + δ a 0 , j ) , where δ x , j = 1 \delta^{x,j}=1 δ x , j = 1 x ≡ j ( m o d k ) x\equiv j\pmod{k} x ≡ j ( mod k )
In particular, the vanishing orders of l X l_X l X q q q
( a 0 + 1 − δ a 0 , j , … , a r + 1 − δ a r , j ) . (a_0+1-\delta^{a_0,j},\ldots,a_r+1-\delta^{a_r,j}). ( a 0 + 1 − δ a 0 , j , … , a r + 1 − δ a r , j ) . Since h 0 ( X , L X ) ≥ r + 2 h^0(X,L_X)\ge r+2 h 0 ( X , L X ) ≥ r + 2 X X X g d r + 1 g^{r+1}_d g d r + 1 q q q
( 0 , a 0 + 1 − δ a 0 , j , … , a r + 1 − δ a r , j ) . (0,a_0+1-\delta^{a_0,j},\ldots,a_r+1-\delta^{a_r,j}). ( 0 , a 0 + 1 − δ a 0 , j , … , a r + 1 − δ a r , j ) . Tensoring with O X ( − q ) \mathcal{O}_X(-q) O X ( − q ) X X X g d − 1 r g^r_{d-1} g d − 1 r q q q
( a 0 − δ a 0 , j , … , a r − δ a r , j ) . (a_0-\delta^{a_0,j},\ldots,a_r-\delta^{a_r,j}). ( a 0 − δ a 0 , j , … , a r − δ a r , j ) . But the r + 1 r+1 r + 1 X X X g d − 1 r g^r_{d-1} g d − 1 r q q q a ′ \mathbf{a}' a ′ a ′ \mathbf{a}' a ′
( a 0 − δ a 0 , j , … , a r − δ a r , j ) (a_0-\delta^{a_0,j},\ldots,a_r-\delta^{a_r,j}) ( a 0 − δ a 0 , j , … , a r − δ a r , j ) on all positions.
This sequence a ′ = ( a 0 ′ , … , a r ′ ) \mathbf{a}'=(a'_0,\ldots,a'_r) a ′ = ( a 0 ′ , … , a r ′ ) a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e )
a 0 ′ = max { 0 , a 0 − δ a 0 , j } , a'_0=\max\{0,a_0-\delta^{a_0,j}\}, a 0 ′ = max { 0 , a 0 − δ a 0 , j } , while for i > 0 i>0 i > 0
a i ′ = { a i − 1 , if a i − 1 ≠ a i − 1 and δ a i , j = 1 , a i , otherwise. a'_i=
\begin{cases}
a_i-1, & \text{if }a_{i-1}\ne a_i-1\text{ and }\delta^{a_i,j}=1,\\
a_i, & \text{otherwise.}
\end{cases} a i ′ = { a i − 1 , a i , if a i − 1 = a i − 1 and δ a i , j = 1 , otherwise. Id est, a vanishing order n ⋅ k + j n\cdot k+j n ⋅ k + j n ⋅ k + j − 1 n\cdot k+j-1 n ⋅ k + j − 1 n ⋅ k + j − 1 n\cdot k+j-1 n ⋅ k + j − 1 n ⋅ k + j − 1 n\cdot k+j-1 n ⋅ k + j − 1 n ⋅ k + j − 1 > e j − 1 ⋅ k + j − 1 n\cdot k+j-1>e_{j-1}\cdot k+j-1 n ⋅ k + j − 1 > e j − 1 ⋅ k + j − 1 a ′ \mathbf{a}' a ′ e j e_j e j e j − 1 e_{j-1} e j − 1 j > 0 j>0 j > 0 e j > e j − 1 e_j>e_{j-1} e j > e j − 1 e k − 1 + 1 e_{k-1}+1 e k − 1 + 1 e 0 − 1 e_0-1 e 0 − 1 j = 0 j=0 j = 0
In total, a ′ = a ( t j ( e ⃗ ) ) \mathbf{a}'=\mathbf{a}(t_j(\vec{e})) a ′ = a ( t j ( e )) ( L X , L E ) (L_X,L_E) ( L X , L E ) g d r g^r_d g d r a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e )
Let now e ⃗ \vec{e} e f ⃗ \vec{f} f k k k [ π : C → P 1 , p ] ∈ H g , k t o t [\pi:C\to\mathbb{P}^1,p]\in\mathcal{H}^{\mathrm{tot}}_{g,k} [ π : C → P 1 , p ] ∈ H g , k tot W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) W f ⃗ ( π , p ) W^{\vec{f}}(\pi,p) W f ( π , p ) Definition 1.2 . We would like these loci to behave well with respect to inclusions: namely if a ( e ⃗ ) ≤ a ( f ⃗ ) \mathbf{a}(\vec{e})\le\mathbf{a}(\vec{f}) a ( e ) ≤ a ( f )
W f ⃗ ( π , p ) ⊆ W e ⃗ ( π , p ) . W^{\vec{f}}(\pi,p)\subseteq W^{\vec{e}}(\pi,p). W f ( π , p ) ⊆ W e ( π , p ) . To compare two pointed Hurwitz–Brill–Noether spaces, we first define a partial order ⪰ \succeq ⪰
Definition 1.13. Let e ⃗ \vec{e} e f ⃗ \vec{f} f ∣ e ⃗ ∣ = ∣ f ⃗ ∣ |\vec{e}|=|\vec{f}| ∣ e ∣ = ∣ f ∣
f ⃗ ⪰ e ⃗ ⟺ ∑ i = 0 n f i ≥ ∑ i = 0 n e i for every 0 ≤ n ≤ k − 1. \vec{f}\succeq\vec{e}\Longleftrightarrow\sum_{i=0}^{n}f_i\ge\sum_{i=0}^{n}e_i\text{ for every }0\le n\le k-1. f ⪰ e ⟺ i = 0 ∑ n f i ≥ i = 0 ∑ n e i for every 0 ≤ n ≤ k − 1. This definition will be useful when working with vectors having ordered entries. As such, we define ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) e ⃗ \vec{e} e
Then, we can relate the entry-wise inequality a ( e ⃗ ) ≤ a ( f ⃗ ) \mathbf{a}(\vec{e})\le\mathbf{a}(\vec{f}) a ( e ) ≤ a ( f )
Lemma 1.14. Let e ⃗ , f ⃗ ∈ Z ≥ 0 k \vec{e},\vec{f}\in\mathbb{Z}^k_{\ge0} e , f ∈ Z ≥ 0 k ∣ e ⃗ ∣ = ∣ f ⃗ ∣ |\vec{e}|=|\vec{f}| ∣ e ∣ = ∣ f ∣ a ( e ⃗ ) ≤ a ( f ⃗ ) \mathbf{a}(\vec{e})\le\mathbf{a}(\vec{f}) a ( e ) ≤ a ( f )
ord ( f ⃗ ) ⪰ ord ( e ⃗ ) . \operatorname{ord}(\vec{f})\succeq\operatorname{ord}(\vec{e}). ord ( f ) ⪰ ord ( e ) . Proof. Since ∣ e ⃗ ∣ = ∣ f ⃗ ∣ \lvert\vec{e}\rvert=\lvert\vec{f}\rvert ∣ e ∣ = ∣ f ∣ x x x ord ( f ⃗ ) \operatorname{ord}(\vec{f}) ord ( f ) x x x ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) 1 ≤ x ≤ k 1\leq x\leq k 1 ≤ x ≤ k
Denote by A i A_i A i B i B_i B i e ⃗ \vec{e} e f ⃗ \vec{f} f i i i A 0 ≤ B 0 A_0\leq B_0 A 0 ≤ B 0 a ⃗ ( e ⃗ ) < 2 k \vec{a}(\vec{e})_{<2k} a ( e ) < 2 k a ⃗ ( f ⃗ ) < 2 k \vec{a}(\vec{f})_{<2k} a ( f ) < 2 k a ⃗ ( e ⃗ ) ≤ a ⃗ ( f ⃗ ) \vec{a}(\vec{e})\leq\vec{a}(\vec{f}) a ( e ) ≤ a ( f ) a ⃗ ( f ⃗ ) < 2 k \vec{a}(\vec{f})_{<2k} a ( f ) < 2 k 1 ≤ x ≤ B 0 1\leq x\leq B_0 1 ≤ x ≤ B 0 x x x ord ( f ⃗ ) \operatorname{ord}(\vec{f}) ord ( f ) ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e )
Because x ≤ B 0 x\leq B_0 x ≤ B 0 x x x ord ( f ⃗ ) \operatorname{ord}(\vec{f}) ord ( f ) 0 0 0 ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) x x x
0 ⋅ min { A 0 , x } + 1 ⋅ ( x − min { A 0 , x } ) . 0\cdot\min\{A_0,x\}+1\cdot\left(x-\min\{A_0,x\}\right). 0 ⋅ min { A 0 , x } + 1 ⋅ ( x − min { A 0 , x } ) . It is clear that x − min { A 0 , x } ≥ 0 x-\min\{A_0,x\}\geq0 x − min { A 0 , x } ≥ 0
Figure 3. This is an example for k = 5 k=5 k = 5 e ⃗ = ( 5 , 4 , 0 , 1 , 0 ) \vec{e}=(5,4,0,1,0) e = ( 5 , 4 , 0 , 1 , 0 ) a ⃗ ( e ⃗ ) \vec{a}(\vec{e}) a ( e ) a ⃗ ( e ⃗ ) < 3 k \vec{a}(\vec{e})_{<3k} a ( e ) < 3 k 3 k 3k 3 k a ⃗ ( e ⃗ ) \vec{a}(\vec{e}) a ( e )
The number of gaps is computed as 2 A 0 + A 1 = 2 ⋅ 2 + 1 2A_0+A_1=2\cdot2+1 2 A 0 + A 1 = 2 ⋅ 2 + 1
Now we prove this more generally. By looking at the number of gaps in a ⃗ ( e ⃗ ) < ( n + 2 ) k \vec{a}(\vec{e})_{<(n+2)k} a ( e ) < ( n + 2 ) k a ⃗ ( f ⃗ ) < ( n + 2 ) k \vec{a}(\vec{f})_{<(n+2)k} a ( f ) < ( n + 2 ) k
( n + 1 ) A 0 + n A 1 + ⋯ + A n ≤ ( n + 1 ) B 0 + n B 1 + ⋯ + B n . (n+1)A_0+nA_1+\cdots+A_n\leq(n+1)B_0+nB_1+\cdots+B_n. ( n + 1 ) A 0 + n A 1 + ⋯ + A n ≤ ( n + 1 ) B 0 + n B 1 + ⋯ + B n . Let 1 ≤ x ≤ B n 1\leq x\leq B_n 1 ≤ x ≤ B n B 0 + B 1 + ⋯ + B n − 1 + x B_0+B_1+\cdots+B_{n-1}+x B 0 + B 1 + ⋯ + B n − 1 + x ord ( f ⃗ ) \operatorname{ord}(\vec{f}) ord ( f ) ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) B 0 + B 1 + ⋯ + B n − 1 + x B_0+B_1+\cdots+B_{n-1}+x B 0 + B 1 + ⋯ + B n − 1 + x ord ( f ⃗ ) \operatorname{ord}(\vec{f}) ord ( f )
B 1 + ⋯ + ( n − 1 ) B n − 1 + n x . B_1+\cdots+(n-1)B_{n-1}+nx. B 1 + ⋯ + ( n − 1 ) B n − 1 + n x . On the other hand, for ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e )
m 1 + 2 m 2 + ⋯ + ( n − 1 ) m n − 1 + n ( B 0 + ⋯ + B n − 1 + x − A 0 − m 1 − ⋯ − m n − 1 ) , m_1+2m_2+\cdots+(n-1)m_{n-1}+n(B_0+\cdots+B_{n-1}+x-A_0-m_1-\cdots-m_{n-1}), m 1 + 2 m 2 + ⋯ + ( n − 1 ) m n − 1 + n ( B 0 + ⋯ + B n − 1 + x − A 0 − m 1 − ⋯ − m n − 1 ) , where m 1 = max { 0 , min { A 1 , B 0 + ⋯ + B n − 1 + x − A 0 } } m_1=\max\{0,\min\{A_1,B_0+\cdots+B_{n-1}+x-A_0\}\} m 1 = max { 0 , min { A 1 , B 0 + ⋯ + B n − 1 + x − A 0 }} m j m_j m j
m j = max { 0 , min { A j , B 0 + ⋯ + B n − 1 + x − A 0 − ∑ i = 1 j − 1 m i } } . m_j=\max\left\{0,\min\left\{A_j,B_0+\cdots+B_{n-1}+x-A_0-\sum_{i=1}^{j-1}m_i\right\}\right\}. m j = max { 0 , min { A j , B 0 + ⋯ + B n − 1 + x − A 0 − i = 1 ∑ j − 1 m i } } . In order to conclude ord ( f ⃗ ) ⪰ ord ( e ⃗ ) \operatorname{ord}(\vec{f})\succeq\operatorname{ord}(\vec{e}) ord ( f ) ⪰ ord ( e ) B 1 + ⋯ + ( n − 1 ) B n − 1 + n x B_1+\cdots+(n-1)B_{n-1}+nx B 1 + ⋯ + ( n − 1 ) B n − 1 + n x
∑ i = 1 n − 1 ( n − i + 1 ) B i − 1 ≥ n A 0 + ∑ i = 1 n − 1 ( n − i ) m i . \sum_{i=1}^{n-1}(n-i+1)B_{i-1}\geq nA_0+\sum_{i=1}^{n-1}(n-i)m_i. i = 1 ∑ n − 1 ( n − i + 1 ) B i − 1 ≥ n A 0 + i = 1 ∑ n − 1 ( n − i ) m i . Using the inequality m i ≤ A i m_i \le A_i m i ≤ A i
n B 0 + ( n − 1 ) B 1 + ⋯ + B n − 1 ≥ n A 0 + ( n − 1 ) A 1 + ⋯ + A n − 1 , nB_0+(n-1)B_1+\cdots+B_{n-1} \ge nA_0+(n-1)A_1+\cdots+A_{n-1}, n B 0 + ( n − 1 ) B 1 + ⋯ + B n − 1 ≥ n A 0 + ( n − 1 ) A 1 + ⋯ + A n − 1 , we deduce our conclusion. Hence, the sum of the last x x x ord ( f ⃗ ) \operatorname{ord}(\vec{f}) ord ( f ) x x x ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e )
x ∈ [ 1 , B 0 ] ∪ [ B 0 + 1 , B 0 + B 1 ] ∪ [ B 0 + B 1 + 1 , B 0 + B 1 + B 2 ] ∪ ⋯ = { 1 , 2 , … , k } . x \in[1,B_0]\cup[B_0+1,B_0+B_1]\cup[B_0+B_1+1,B_0+B_1+B_2]\cup\cdots=\{1,2,\ldots,k\}. x ∈ [ 1 , B 0 ] ∪ [ B 0 + 1 , B 0 + B 1 ] ∪ [ B 0 + B 1 + 1 , B 0 + B 1 + B 2 ] ∪ ⋯ = { 1 , 2 , … , k } . This is exactly what we wanted to prove. □
Let e ⃗ = ( e 0 , … , e k − 1 ) \vec{e}=(e_0,\ldots,e_{k-1}) e = ( e 0 , … , e k − 1 ) dim W e ⃗ ( π , p ) \dim W^{\vec{e}}(\pi,p) dim W e ( π , p )
Definition 1.15. We define the functions
u ( e ⃗ ) : = ∑ e i < e j ( e j − e i − 1 ) u(\vec{e}) := \sum_{e_i<e_j}(e_j-e_i-1) u ( e ) := e i < e j ∑ ( e j − e i − 1 ) and
inv ( e ⃗ ) : = # { ( i , j ) ∣ 0 ≤ i < j ≤ k − 1 and e i < e j } . \operatorname{inv}(\vec{e}) := \#\{(i,j)\mid0\le i<j\le k-1\text{ and }e_i<e_j\}. inv ( e ) := # {( i , j ) ∣ 0 ≤ i < j ≤ k − 1 and e i < e j } . Notice that our definition differs from the standard way the number of inversions is defined. This is because dim W e ⃗ ( π , p ) \dim W^{\vec{e}}(\pi,p) dim W e ( π , p )
By degenerating to the boundary, computing the upper-bound for the dimension will reduce to high-school level combinatorics. We first present the result in the form in which it was proposed to the final competition of the International Mathematical Summer Camp 2025, and then translate it back into the language of Brill–Noether Theory.
Proposition 1.16. Let e ⃗ = ( e 0 , e 1 , … , e k − 1 ) \vec{e}=(e_0,e_1,\ldots,e_{k-1}) e = ( e 0 , e 1 , … , e k − 1 ) N N N N > max { e i ∣ 0 ≤ i ≤ k − 1 } N>\max\{e_i\mid0\le i\le k-1\} N > max { e i ∣ 0 ≤ i ≤ k − 1 } N × k N\times k N × k 0 , 1 , … , k N − 1 0,1,\ldots,kN-1 0 , 1 , … , k N − 1 0 ≤ i ≤ k − 1 0\le i\le k-1 0 ≤ i ≤ k − 1 e i + 1 e_i+1 e i + 1 i i i
A move consists in choosing a column, and trying to move all the coins in that column to the squares numbered one less, i.e.
Then we always need to perform exactly
u ( e ⃗ ) + inv ( e ⃗ ) u(\vec{e})+\operatorname{inv}(\vec{e}) u ( e ) + inv ( e ) nontrivial moves in order to arrange the coins on the squares numbered 0 , 1 , … , k − 1 + ∑ i = 0 k − 1 e i 0,1,\ldots,k-1+\sum_{i=0}^{k-1}e_i 0 , 1 , … , k − 1 + ∑ i = 0 k − 1 e i
Proof. The proof is a simple induction on the value u ( e ⃗ ) + inv ( e ⃗ ) u(\vec{e})+\operatorname{inv}(\vec{e}) u ( e ) + inv ( e ) 0 0 0 0 , 1 , … , k − 1 + ∑ i = 0 k − 1 e i 0,1,\ldots,k-1+\sum_{i=0}^{k-1}e_i 0 , 1 , … , k − 1 + ∑ i = 0 k − 1 e i u ( e ⃗ ) + inv ( e ⃗ ) u(\vec{e})+\operatorname{inv}(\vec{e}) u ( e ) + inv ( e )
When u ( e ⃗ ) + inv ( e ⃗ ) > 0 u(\vec{e})+\operatorname{inv}(\vec{e})>0 u ( e ) + inv ( e ) > 0
Case 1: If we choose the column i i i 1 ≤ i ≤ k − 1 1\le i\le k-1 1 ≤ i ≤ k − 1
t i ( e ⃗ ) : = ( e 0 , e 1 , … , e i − 2 , e i , e i − 1 , … , e k − 1 ) . t_i(\vec{e}):=(e_0,e_1,\ldots,e_{i-2},e_i,e_{i-1},\ldots,e_{k-1}). t i ( e ) := ( e 0 , e 1 , … , e i − 2 , e i , e i − 1 , … , e k − 1 ) . We have u ( e ⃗ ) = u ( t i ( e ⃗ ) ) u(\vec{e})=u(t_i(\vec{e})) u ( e ) = u ( t i ( e ))
Case 2: We choose column 0 0 0 k − 1 k-1 k − 1
t 0 ( e ⃗ ) : = ( e k − 1 + 1 , e 1 , e 2 , … , e k − 2 , e 0 − 1 ) t_0(\vec{e}):=(e_{k-1}+1,e_1,e_2,\ldots,e_{k-2},e_0-1) t 0 ( e ) := ( e k − 1 + 1 , e 1 , e 2 , … , e k − 2 , e 0 − 1 ) and we immediately check the equality
u ( e ⃗ ) − u ( t 0 ( e ⃗ ) ) = # { 1 ≤ j ≤ k − 2 ∣ e j + 1 < e 0 } + # { 1 ≤ j ≤ k − 2 ∣ e j − 1 > e k − 1 } − ⋯ u(\vec{e})-u(t_0(\vec{e}))=\#\{1\leq j\leq k-2\mid e_j+1<e_0\}+\#\{1\leq j\leq k-2\mid e_j-1>e_{k-1}\}-\cdots u ( e ) − u ( t 0 ( e )) = # { 1 ≤ j ≤ k − 2 ∣ e j + 1 < e 0 } + # { 1 ≤ j ≤ k − 2 ∣ e j − 1 > e k − 1 } − ⋯ − # { 1 ≤ j ≤ k − 2 ∣ e j > e 0 } − # { 1 ≤ j ≤ k − 2 ∣ e j < e k − 1 } + ⋯ + e 0 − e k − 1 − 1 − max { 0 , e 0 − e k − 1 − 3 } . \begin{aligned}
-\#\{1\leq j\leq k-2\mid e_j>e_0\}-\#\{1\leq j\leq k-2\mid e_j<e_{k-1}\}+\cdots\\
&+e_0-e_{k-1}-1-\max\{0,e_0-e_{k-1}-3\}.
\end{aligned} − # { 1 ≤ j ≤ k − 2 ∣ e j > e 0 } − # { 1 ≤ j ≤ k − 2 ∣ e j < e k − 1 } + ⋯ + e 0 − e k − 1 − 1 − max { 0 , e 0 − e k − 1 − 3 } . To see this, we look at the contribution to u ( e ⃗ ) u(\vec{e}) u ( e ) ( e 0 , e j ) (e_0,e_j) ( e 0 , e j ) ( e k − 1 , e j ) (e_{k-1},e_j) ( e k − 1 , e j ) e 0 e_0 e 0 e k − 1 e_{k-1} e k − 1
On the other hand, a similar argument implies
inv ( e ⃗ ) − inv ( t 0 ( e ⃗ ) ) = # { 1 ≤ j ≤ k − 2 ∣ e j > e 0 } + # { 1 ≤ j ≤ k − 2 ∣ e j < e k − 1 } − ⋯ \operatorname{inv}(\vec{e})-\operatorname{inv}(t_0(\vec{e}))=\#\{1\leq j\leq k-2\mid e_j>e_0\}+\#\{1\leq j\leq k-2\mid e_j<e_{k-1}\}-\cdots inv ( e ) − inv ( t 0 ( e )) = # { 1 ≤ j ≤ k − 2 ∣ e j > e 0 } + # { 1 ≤ j ≤ k − 2 ∣ e j < e k − 1 } − ⋯ − # { 1 ≤ j ≤ k − 2 ∣ e j < e 0 − 1 } − # { 1 ≤ j ≤ k − 2 ∣ e j > e k − 1 + 1 } − ⋯ -\#\{1\leq j\leq k-2\mid e_j<e_0-1\}-\#\{1\leq j\leq k-2\mid e_j>e_{k-1}+1\}-\cdots − # { 1 ≤ j ≤ k − 2 ∣ e j < e 0 − 1 } − # { 1 ≤ j ≤ k − 2 ∣ e j > e k − 1 + 1 } − ⋯ − { 1 if e 0 − e k − 1 ≥ 3 , 0 if e 0 − e k − 1 = 2. -\begin{cases}
1 & \text{if }e_0-e_{k-1}\geq3,\\
0 & \text{if }e_0-e_{k-1}=2.
\end{cases} − { 1 0 if e 0 − e k − 1 ≥ 3 , if e 0 − e k − 1 = 2. Adding the two equalities together, we obtain
u ( t 0 ( e ⃗ ) ) + inv ( t 0 ( e ⃗ ) ) = u ( e ⃗ ) + inv ( e ⃗ ) − 1. u(t_0(\vec{e}))+\operatorname{inv}(t_0(\vec{e}))=u(\vec{e})+\operatorname{inv}(\vec{e})-1. u ( t 0 ( e )) + inv ( t 0 ( e )) = u ( e ) + inv ( e ) − 1. This means that each possible move reduces this semi-invariant by exactly 1, and hence we need exactly u ( e ⃗ ) + inv ( e ⃗ ) u(\vec{e})+\operatorname{inv}(\vec{e}) u ( e ) + inv ( e )
0 , 1 , … , k − 1 + ∣ e ⃗ ∣ 0,1,\ldots,k-1+|\vec{e}| 0 , 1 , … , k − 1 + ∣ e ∣ i.e. until the coins are in the unique position where no further moves are possible. □
There is another way in which we can interpret the semi-invariant u ( e ⃗ ) + inv ( e ⃗ ) u(\vec{e})+\operatorname{inv}(\vec{e}) u ( e ) + inv ( e )
Figure 1 (PDF p. 12). Initial position; Final position; Displacement number.
For e ⃗ = ( 5 , 4 , 0 , 1 , 0 ) \vec{e}=(5,4,0,1,0) e = ( 5 , 4 , 0 , 1 , 0 ) u ( e ⃗ ) = 19 u(\vec{e})=19 u ( e ) = 19 inv ( e ⃗ ) = 1 \operatorname{inv}(\vec{e})=1 inv ( e ) = 1
u ( e ⃗ ) + inv ( e ⃗ ) = 11 + 8 + 0 + 1 + 0. u(\vec{e})+\operatorname{inv}(\vec{e})=11+8+0+1+0. u ( e ) + inv ( e ) = 11 + 8 + 0 + 1 + 0. The proof of this statement is identical to the proof of Proposition 1.16 : this semi-invariant decreases by exactly one after each move.
One of our goals is to prove that, if we have the entry-wise inequality a ( e ⃗ ) ≤ a ( f ⃗ ) \mathbf{a}(\vec{e}) \leq\mathbf{a}(\vec{f}) a ( e ) ≤ a ( f ) W ‾ f ⃗ ( π , p ) ⊆ W ‾ e ⃗ ( π , p ) \overline{W}^{\vec{f}}(\pi,p) \subseteq\overline{W}^{\vec{e}}(\pi,p) W f ( π , p ) ⊆ W e ( π , p ) Theorem 2.5 , we will need just an inequality of expected dimensions, derived from the next proposition.
Proposition 1.18. Let e ⃗ , f ⃗ ∈ Z ≥ 0 k \vec{e},\vec{f} \in\mathbb{Z}_{\geq0}^{k} e , f ∈ Z ≥ 0 k ∣ e ⃗ ∣ = ∣ f ⃗ ∣ |\vec{e}|=|\vec{f}| ∣ e ∣ = ∣ f ∣ a ( e ⃗ ) ≤ a ( f ⃗ ) \mathbf{a}(\vec{e}) \leq\mathbf{a}(\vec{f}) a ( e ) ≤ a ( f )
u ( e ⃗ ) + inv ( e ⃗ ) ≤ u ( f ⃗ ) + inv ( f ⃗ ) . u(\vec{e})+\operatorname{inv}(\vec{e}) \leq u(\vec{f})+\operatorname{inv}(\vec{f}). u ( e ) + inv ( e ) ≤ u ( f ) + inv ( f ) . Proof. We will induct on the value s = ∣ e ⃗ ∣ = ∣ f ⃗ ∣ s=|\vec{e}|=|\vec{f}| s = ∣ e ∣ = ∣ f ∣ s = 0 s=0 s = 0 e ⃗ = f ⃗ = ( 0 , … , 0 ) \vec{e}=\vec{f}=(0,\ldots,0) e = f = ( 0 , … , 0 )
Assume s s s e ⃗ , f ⃗ ∈ Z ≥ 0 k \vec{e},\vec{f}\in\mathbb{Z}_{\geq0}^{k} e , f ∈ Z ≥ 0 k
s = ∣ e ⃗ ∣ = ∣ f ⃗ ∣ , a ( e ⃗ ) ≤ a ( f ⃗ ) entry-wise , and u ( e ⃗ ) + inv ( e ⃗ ) > u ( f ⃗ ) + inv ( f ⃗ ) . s=|\vec{e}|=|\vec{f}|,\quad\mathbf{a}(\vec{e})\leq\mathbf{a}(\vec{f})\text{ entry-wise},\quad\text{and}\quad u(\vec{e})+\operatorname{inv}(\vec{e})>u(\vec{f})+\operatorname{inv}(\vec{f}). s = ∣ e ∣ = ∣ f ∣ , a ( e ) ≤ a ( f ) entry-wise , and u ( e ) + inv ( e ) > u ( f ) + inv ( f ) . Moreover, we choose such vectors e ⃗ , f ⃗ \vec{e},\vec{f} e , f u ( f ⃗ ) + inv ( f ⃗ ) u(\vec{f})+\operatorname{inv}(\vec{f}) u ( f ) + inv ( f )
Proof of Claim 1. If f 0 ≠ 0 f_0\neq0 f 0 = 0 k + 1 k+1 k + 1 a ( f ⃗ ) \mathbf{a}(\vec{f}) a ( f ) ( 0 , 1 , … , k ) (0,1,\ldots,k) ( 0 , 1 , … , k ) a ( e ⃗ ) ≤ a ( f ⃗ ) \mathbf{a}(\vec{e})\leq\mathbf{a}(\vec{f}) a ( e ) ≤ a ( f ) k + 1 k+1 k + 1 a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) ( 0 , 1 , … , k ) (0,1,\ldots,k) ( 0 , 1 , … , k ) e 0 ≠ 0 e_0\neq0 e 0 = 0
Taking σ ( e ⃗ ) = ( e 1 , … , e k − 1 , e 0 − 1 ) \sigma(\vec{e})=(e_1,\ldots,e_{k-1},e_0-1) σ ( e ) = ( e 1 , … , e k − 1 , e 0 − 1 ) σ ( f ⃗ ) = ( f 1 , … , f k − 1 , f 0 − 1 ) \sigma(\vec{f})=(f_1,\ldots,f_{k-1},f_0-1) σ ( f ) = ( f 1 , … , f k − 1 , f 0 − 1 ) a ( σ ( e ⃗ ) ) ≤ a ( σ ( f ⃗ ) ) \mathbf{a}(\sigma(\vec{e}))\leq\mathbf{a}(\sigma(\vec{f})) a ( σ ( e )) ≤ a ( σ ( f )) a ( σ ( e ⃗ ) ) \mathbf{a}(\sigma(\vec{e})) a ( σ ( e )) a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) 0 0 0 1 1 1
The minimality of s s s
u ( σ ( e ⃗ ) ) + inv ( σ ( e ⃗ ) ) ≤ u ( σ ( f ⃗ ) ) + inv ( σ ( f ⃗ ) ) . u(\sigma(\vec{e}))+\operatorname{inv}(\sigma(\vec{e}))\leq u(\sigma(\vec{f}))+\operatorname{inv}(\sigma(\vec{f})). u ( σ ( e )) + inv ( σ ( e )) ≤ u ( σ ( f )) + inv ( σ ( f )) . But the invariant u + inv u+\operatorname{inv} u + inv Remark 1.17 . The process of removing 0 0 0 1 1 1
⇒ u ( e ⃗ ) + inv ( e ⃗ ) = u ( σ ( e ⃗ ) ) + inv ( σ ( e ⃗ ) ) ≤ u ( σ ( f ⃗ ) ) + inv ( σ ( f ⃗ ) ) = u ( f ⃗ ) + inv ( f ⃗ ) , \Rightarrow u(\vec{e})+\operatorname{inv}(\vec{e})=u(\sigma(\vec{e}))+\operatorname{inv}(\sigma(\vec{e}))\leq u(\sigma(\vec{f}))+\operatorname{inv}(\sigma(\vec{f}))=u(\vec{f})+\operatorname{inv}(\vec{f}), ⇒ u ( e ) + inv ( e ) = u ( σ ( e )) + inv ( σ ( e )) ≤ u ( σ ( f )) + inv ( σ ( f )) = u ( f ) + inv ( f ) , contradicting the choice of e ⃗ \vec{e} e f ⃗ \vec{f} f
Hence we must have f 0 = 0 f_0=0 f 0 = 0 f i f_i f i f ⃗ \vec{f} f m : = min { e i , f i } m:=\min\{e_i,f_i\} m := min { e i , f i } i , k + i , … , m ⋅ k + i i,k+i,\ldots,m\cdot k+i i , k + i , … , m ⋅ k + i m m m a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) a ( f ⃗ ) \mathbf{a}(\vec{f}) a ( f )
For each 0 ≤ x ≤ m 0\leq x\leq m 0 ≤ x ≤ m N ( x , e ⃗ ) N(x,\vec{e}) N ( x , e ) i + x ⋅ k i+x\cdot k i + x ⋅ k a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) N ( x , f ⃗ ) N(x,\vec{f}) N ( x , f ) i + x ⋅ k i+x\cdot k i + x ⋅ k a ( f ⃗ ) \mathbf{a}(\vec{f}) a ( f )
Claim 2. For each 0 ≤ x ≤ m 0\leq x\leq m 0 ≤ x ≤ m N ( x , e ⃗ ) ≥ N ( x , f ⃗ ) N(x,\vec{e})\geq N(x,\vec{f}) N ( x , e ) ≥ N ( x , f )
Proof of Claim 2. If N ( x , e ⃗ ) < N ( x , f ⃗ ) N(x,\vec{e})<N(x,\vec{f}) N ( x , e ) < N ( x , f ) a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) N ( x , f ⃗ ) N(x,\vec{f}) N ( x , f )
i + x ⋅ k + N ( x , f ⃗ ) − N ( x , e ⃗ ) . i+x\cdot k+N(x,\vec{f})-N(x,\vec{e}). i + x ⋅ k + N ( x , f ) − N ( x , e ) . In this case, the entry-wise inequality a ( e ⃗ ) ≤ a ( f ⃗ ) \mathbf{a}(\vec{e})\leq\mathbf{a}(\vec{f}) a ( e ) ≤ a ( f ) N ( x , f ⃗ ) N(x,\vec{f}) N ( x , f )
Let Δ ( x ) : = N ( x , e ⃗ ) − N ( x , f ⃗ ) \Delta(x):=N(x,\vec{e})-N(x,\vec{f}) Δ ( x ) := N ( x , e ) − N ( x , f )
Claim 3. There exists 1 ≤ x ≤ m 1 \le x \le m 1 ≤ x ≤ m Δ ( x ) = 0 \Delta(x)=0 Δ ( x ) = 0
Proof. If Δ ( x ) ≠ 0 \Delta(x) \ne0 Δ ( x ) = 0 1 ≤ x ≤ m 1 \le x \le m 1 ≤ x ≤ m i + x ⋅ k i+x\cdot k i + x ⋅ k a ( f ⃗ ) \mathbf{a}(\vec{f}) a ( f ) a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e )
Moreover, since for m < x ≤ f i m<x\le f_i m < x ≤ f i i + x ⋅ k i+x\cdot k i + x ⋅ k a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) i + x ⋅ k i+x\cdot k i + x ⋅ k a ( f ⃗ ) \mathbf{a}(\vec{f}) a ( f ) a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e )
Hence, reducing all entries i + x ⋅ k i+x\cdot k i + x ⋅ k 1 ≤ x ≤ f i 1\le x\le f_i 1 ≤ x ≤ f i a ( f ⃗ ) \mathbf{a}(\vec{f}) a ( f ) a ( e ⃗ ) ≤ a ( t i ( f ⃗ ) ) \mathbf{a}(\vec{e})\le\mathbf{a}(t_i(\vec{f})) a ( e ) ≤ a ( t i ( f ))
But we know
u ( t i ( f ⃗ ) ) + inv ( t i ( f ⃗ ) ) = u ( f ⃗ ) + inv ( f ⃗ ) − 1 u(t_i(\vec{f}))+\operatorname{inv}(t_i(\vec{f}))=u(\vec{f})+\operatorname{inv}(\vec{f})-1 u ( t i ( f )) + inv ( t i ( f )) = u ( f ) + inv ( f ) − 1 and the assumption on minimality of u ( f ⃗ ) + inv ( f ⃗ ) u(\vec{f})+\operatorname{inv}(\vec{f}) u ( f ) + inv ( f )
u ( e ⃗ ) + inv ( e ⃗ ) ≤ u ( f ⃗ ) + inv ( f ⃗ ) − 1. u(\vec{e})+\operatorname{inv}(\vec{e})\le u(\vec{f})+\operatorname{inv}(\vec{f})-1. u ( e ) + inv ( e ) ≤ u ( f ) + inv ( f ) − 1. This contradicts the assumption, and hence the claim follows.
Hence there exists values of x x x m m m Δ ( x ) = 0 \Delta(x)=0 Δ ( x ) = 0 x x x a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) a ( f ⃗ ) \mathbf{a}(\vec{f}) a ( f ) i + x ⋅ k i+x\cdot k i + x ⋅ k N ( x , e ⃗ ) = N ( x , f ⃗ ) N(x,\vec{e})=N(x,\vec{f}) N ( x , e ) = N ( x , f )
We look at the entry N ( x , f ⃗ ) − 1 N(x,\vec{f})-1 N ( x , f ) − 1 f i − 1 = 0 f_{i-1}=0 f i − 1 = 0 a ( f ⃗ ) \mathbf{a}(\vec{f}) a ( f ) N ( x , f ⃗ ) − 1 N(x,\vec{f})-1 N ( x , f ) − 1 i − 1 + x ⋅ k i-1+x\cdot k i − 1 + x ⋅ k a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) N ( x , f ⃗ ) − 1 N(x,\vec{f})-1 N ( x , f ) − 1 i − 1 + x ⋅ k i-1+x\cdot k i − 1 + x ⋅ k
This immediately implies the entry-wise inequality
a ( t i ( e ⃗ ) ) ≤ a ( t i ( f ⃗ ) ) : \mathbf{a}(t_i(\vec{e}))\le\mathbf{a}(t_i(\vec{f})): a ( t i ( e )) ≤ a ( t i ( f )) : When Δ ( x ) ≠ 0 \Delta(x)\ne0 Δ ( x ) = 0 x > m x>m x > m Δ ( x ) = 0 \Delta(x)=0 Δ ( x ) = 0 e i − 1 e_{i-1} e i − 1 e i e_i e i f i − 1 f_{i-1} f i − 1 f i f_i f i N ( x , e ⃗ ) = N ( x , f ⃗ ) N(x,\vec{e})=N(x,\vec{f}) N ( x , e ) = N ( x , f )
The minimality of u ( f ⃗ ) + inv ( f ⃗ ) u(\vec{f})+\operatorname{inv}(\vec{f}) u ( f ) + inv ( f )
u ( t i ( e ⃗ ) ) + inv ( t i ( e ⃗ ) ) = u ( e ⃗ ) + inv ( e ⃗ ) − 1 ≤ u ( t i ( f ⃗ ) ) + inv ( t i ( f ⃗ ) ) = u ( f ⃗ ) + inv ( f ⃗ ) − 1. u(t_i(\vec{e}))+\operatorname{inv}(t_i(\vec{e}))=u(\vec{e})+\operatorname{inv}(\vec{e})-1\le u(t_i(\vec{f}))+\operatorname{inv}(t_i(\vec{f}))=u(\vec{f})+\operatorname{inv}(\vec{f})-1. u ( t i ( e )) + inv ( t i ( e )) = u ( e ) + inv ( e ) − 1 ≤ u ( t i ( f )) + inv ( t i ( f )) = u ( f ) + inv ( f ) − 1. This contradicts the assumption on e ⃗ \vec{e} e f ⃗ \vec{f} f e ⃗ \vec{e} e f ⃗ \vec{f} f
∣ e ⃗ ∣ = ∣ f ⃗ ∣ , a ( e ⃗ ) ≤ a ( f ⃗ ) entry-wise and u ( e ⃗ ) + inv ( e ⃗ ) > u ( f ⃗ ) + inv ( f ⃗ ) . |\vec{e}|=|\vec{f}|,\quad\mathbf{a}(\vec{e})\le\mathbf{a}(\vec{f})\text{ entry-wise and }u(\vec{e})+\operatorname{inv}(\vec{e})>u(\vec{f})+\operatorname{inv}(\vec{f}). ∣ e ∣ = ∣ f ∣ , a ( e ) ≤ a ( f ) entry-wise and u ( e ) + inv ( e ) > u ( f ) + inv ( f ) . Moreover, the same inductive method implies that if u ( e ⃗ ) + inv ( e ⃗ ) = u ( f ⃗ ) + inv ( f ⃗ ) u(\vec{e})+\operatorname{inv}(\vec{e})=u(\vec{f})+\operatorname{inv}(\vec{f}) u ( e ) + inv ( e ) = u ( f ) + inv ( f ) e ⃗ = f ⃗ \vec{e}=\vec{f} e = f
Dimension results
The goal of this section is to compute the dimension of pointed Hurwitz–Brill–Noether loci W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) [ π : C → P 1 , p ] ∈ H g , k t o t [\pi:C\to\mathbb{P}^{1},p]\in\mathcal{H}^{\mathrm{tot}}_{g,k} [ π : C → P 1 , p ] ∈ H g , k tot
dim W e ⃗ ( π , p ) = g − u ( e ⃗ ) − inv ( e ⃗ ) \dim W^{\vec{e}}(\pi,p)=g-u(\vec{e})-\operatorname{inv}(\vec{e}) dim W e ( π , p ) = g − u ( e ) − inv ( e ) whenever this value is non-negative, and the locus is empty otherwise.
The upper bound dim W e ⃗ ( π , p ) ≤ g − u ( e ⃗ ) − inv ( e ⃗ ) \dim W^{\vec{e}}(\pi,p)\le g-u(\vec{e})-\operatorname{inv}(\vec{e}) dim W e ( π , p ) ≤ g − u ( e ) − inv ( e ) H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot Lemma 1.12 . Proposition 1.18 together with an induction argument will then imply our inequality.
To obtain a lower-bound on the dimension, we realize W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) [15 ] and uses e ⃗ \vec{e} e Proposition 1.6 to control the ramification orders and realizes the pointed Hurwitz–Brill–Noether locus as an intersection of flag degeneracy loci. This second proof is independent of [15 ] .
Proposition 2.1. Let [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^{1},p] [ π : C → P 1 , p ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot e ⃗ = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k \vec{e}=(e_{0},\ldots,e_{k-1})\in\mathbb{Z}_{\geq0}^{k} e = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k r = k + ∣ e ⃗ ∣ − 1 r=k+|\vec{e}|-1 r = k + ∣ e ∣ − 1 d = g + r d=g+r d = g + r g d r g^{r}_{d} g d r C C C a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) p p p g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec{e})-\operatorname{inv}(\vec{e}) g − u ( e ) − inv ( e )
dim W e ⃗ ( π , p ) ≤ g − u ( e ⃗ ) − inv ( e ⃗ ) . \dim W^{\vec{e}}(\pi,p)\leq g-u(\vec{e})-\operatorname{inv}(\vec{e}). dim W e ( π , p ) ≤ g − u ( e ) − inv ( e ) . In particular, when the expected dimension is negative, the pointed Hurwitz–Brill–Noether locus is the empty set.
Proof. We induct on the genus. When C C C
For genus g ≥ 2 g\geq2 g ≥ 2 [ π 0 : X ∪ q E → P X ∪ q P E , p ] [\pi_{0}:X\cup_{q}E\to P_{X}\cup_{q}P_{E},p] [ π 0 : X ∪ q E → P X ∪ q P E , p ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot [1 ] .
We describe the limit g d r g^{r}_{d} g d r a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) [ X ∪ q E , p ] [X\cup_{q}E,p] [ X ∪ q E , p ] Lemma 1.12 . Let ( L X , L E ) ∈ Pic d ( X ) × Pic d ( E ) (L_{X},L_{E})\in\operatorname{Pic}^{d}(X)\times\operatorname{Pic}^{d}(E) ( L X , L E ) ∈ Pic d ( X ) × Pic d ( E ) ( l X , l E ) (l_{X},l_{E}) ( l X , l E )
If L E L_{E} L E O E ( i ⋅ p + ( d − i ) ⋅ q ) \mathcal{O}_{E}(i\cdot p+(d-i)\cdot q) O E ( i ⋅ p + ( d − i ) ⋅ q ) l X l_{X} l X q q q 1 + a ( e ⃗ ) 1+\mathbf{a}(\vec{e}) 1 + a ( e ) Lemma 1.12 , l X ( − q ) l_{X}(-q) l X ( − q ) g d − 1 r g^{r}_{d-1} g d − 1 r X X X a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) q q q Proposition 1.18 , we obtain that ( L X , L E ) (L_{X},L_{E}) ( L X , L E )
( O X ( q ) ⊗ W d − 1 , a ( e ⃗ ) r ( X , q ) ) × Pic d ( E ) . \left(\mathcal{O}_{X}(q)\otimes W^{r}_{d-1,\mathbf{a}(\vec{e})}(X,q)\right)\times\operatorname{Pic}^{d}(E). ( O X ( q ) ⊗ W d − 1 , a ( e ) r ( X , q ) ) × Pic d ( E ) . Using induction and Proposition 1.18 , we find that all components of W d − 1 , a ( e ⃗ ) r ( X , q ) W^{r}_{d-1,\mathbf{a}(\vec{e})}(X,q) W d − 1 , a ( e ) r ( X , q ) g − 1 − u ( e ⃗ ) − inv ( e ⃗ ) g-1-u(\vec{e})-\operatorname{inv}(\vec{e}) g − 1 − u ( e ) − inv ( e )
( O X ( q ) ⊗ W d − 1 , a ( e ⃗ ) r ( X , q ) ) × Pic d ( E ) \left(\mathcal{O}_{X}(q)\otimes W^{r}_{d-1,\mathbf{a}(\vec{e})}(X,q)\right)\times\operatorname{Pic}^{d}(E) ( O X ( q ) ⊗ W d − 1 , a ( e ) r ( X , q ) ) × Pic d ( E ) has dimension at most g − 1 − u ( e ⃗ ) − inv ( e ⃗ ) + 1 g-1-u(\vec{e})-\operatorname{inv}(\vec{e})+1 g − 1 − u ( e ) − inv ( e ) + 1
If L E ≅ O E ( i ⋅ p + ( d − i ) ⋅ q ) L_{E}\cong\mathcal{O}_{E}(i\cdot p+(d-i)\cdot q) L E ≅ O E ( i ⋅ p + ( d − i ) ⋅ q ) Lemma 1.12 that L X ( − q ) L_{X}(-q) L X ( − q ) W d − 1 , a ( t i ( e ⃗ ) ) r ( X , q ) W^{r}_{d-1,\mathbf{a}(t_{i}(\vec{e}))}(X,q) W d − 1 , a ( t i ( e )) r ( X , q )
In particular, the locus of such pairs ( L X , L E ) (L_{X},L_{E}) ( L X , L E )
( O X ( q ) ⊗ W d − 1 , a ( t i ( e ⃗ ) ) r ( X , q ) ) × { O E ( i ⋅ p + ( d − i ) ⋅ q ) } . \left(\mathcal{O}_{X}(q)\otimes W^{r}_{d-1,\mathbf{a}(t_{i}(\vec{e}))}(X,q)\right)\times\{\mathcal{O}_{E}(i\cdot p+(d-i)\cdot q)\}. ( O X ( q ) ⊗ W d − 1 , a ( t i ( e )) r ( X , q ) ) × { O E ( i ⋅ p + ( d − i ) ⋅ q )} . If we order the ramification orders in columns modulo k k k Proposition 1.16 , we see that a ( t i ( e ⃗ ) ) \mathbf{a}(t_{i}(\vec{e})) a ( t i ( e )) a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) i i i k k k u ( t i ( e ⃗ ) ) + inv ( t i ( e ⃗ ) ) = u ( e ⃗ ) + inv ( e ⃗ ) − 1 u(t_{i}(\vec{e}))+\operatorname{inv}(t_{i}(\vec{e}))=u(\vec{e})+\operatorname{inv}(\vec{e})-1 u ( t i ( e )) + inv ( t i ( e )) = u ( e ) + inv ( e ) − 1 Proposition 1.16 . In particular, using Proposition 1.18 and the induction hypothesis, the locus of limit g d r g^{r}_{d} g d r a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) p p p g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec{e})-\operatorname{inv}(\vec{e}) g − u ( e ) − inv ( e )
dim W d , a ( e ⃗ ) r ( C , p ) ≤ g − u ( e ⃗ ) − inv ( e ⃗ ) . \dim W^{r}_{d,\mathbf{a}(\vec{e})}(C,p)\leq g-u(\vec{e})-\operatorname{inv}(\vec{e}). dim W d , a ( e ) r ( C , p ) ≤ g − u ( e ) − inv ( e ) . Our next goal is to prove the converse inequality.
Proposition 2.2. Let [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^{1},p] [ π : C → P 1 , p ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot e ⃗ = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k \vec{e}=(e_{0},\ldots,e_{k-1})\in\mathbb{Z}^{k}_{\geq0} e = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k r = k + ∣ e ⃗ ∣ − 1 r=k+|\vec{e}|-1 r = k + ∣ e ∣ − 1 d = g + r d=g+r d = g + r g d r g^{r}_{d} g d r C C C a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) p p p
g − u ( e ⃗ ) − inv ( e ⃗ ) , g-u(\vec{e})-\operatorname{inv}(\vec{e}), g − u ( e ) − inv ( e ) , that is dim W e ⃗ ( π , p ) ≥ g − u ( e ⃗ ) − inv ( e ⃗ ) \dim W^{\vec{e}}(\pi,p)\geq g-u(\vec{e})-\operatorname{inv}(\vec{e}) dim W e ( π , p ) ≥ g − u ( e ) − inv ( e )
We will provide two proofs of this result. Prior to the first proof, we recall some useful definitions of nested linear series from [15 Section 6] and set some notation we will use. Our definitions differ very slightly as we retain the ordering of e ⃗ \vec{e} e
Given e ⃗ ∈ Z k \vec{e}\in\mathbb{Z}^{k} e ∈ Z k d 1 > ⋯ > d s ∈ Z d_{1}>\cdots>d_{s}\in\mathbb{Z} d 1 > ⋯ > d s ∈ Z e ⃗ \vec{e} e
M j : = { i ∈ [ 0 , k − 1 ] ∣ e i = d j } M_{j}:=\{i\in[0,k-1]\mid e_{i}=d_{j}\} M j := { i ∈ [ 0 , k − 1 ] ∣ e i = d j } record the positions of e ⃗ \vec{e} e d j d_j d j k j = ∣ M j ∣ k_j=|M_j| k j = ∣ M j ∣ d j d_j d j e ⃗ \vec{e} e M j M_j M j
M j = { m j , 1 < m j , 2 < ⋯ < m j , k j } . M_j=\{m_{j,1}<m_{j,2}<\cdots<m_{j,k_j}\}. M j = { m j , 1 < m j , 2 < ⋯ < m j , k j } . Let π : C → P 1 \pi:C\to\mathbb{P}^1 π : C → P 1 k k k L L L C C C
L ( t ) : = L ⊗ π ∗ O P 1 ( t ) . L(t):=L\otimes\pi^*\mathcal{O}_{\mathbb{P}^1}(t). L ( t ) := L ⊗ π ∗ O P 1 ( t ) . As observed in [15 Section 6] , the condition that π ∗ L \pi_*L π ∗ L e ⃗ \vec{e} e H 0 ( C , L ( − d j ) ) H^0(C,L(-d_j)) H 0 ( C , L ( − d j )) P 1 \mathbb{P}^1 P 1
Definition 2.3. An e ⃗ \vec{e} e C C C ( L , V 1 , … , V s ) (L,V_1,\ldots,V_s) ( L , V 1 , … , V s ) L L L C C C
V j ⊆ H 0 ( C , L ( − d j ) ) , dim V j = h 0 ( P 1 , O P 1 ( e ⃗ ) ( − d j ) ) V_j\subseteq H^0(C,L(-d_j)),\qquad\dim V_j=h^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(\vec{e})(-d_j)) V j ⊆ H 0 ( C , L ( − d j )) , dim V j = h 0 ( P 1 , O P 1 ( e ) ( − d j )) for every 1 ≤ j ≤ s 1\leq j\leq s 1 ≤ j ≤ s 2 ≤ j ≤ s 2\leq j\leq s 2 ≤ j ≤ s
V j − 1 ⊗ H 0 ( P 1 , O P 1 ( d j − 1 − d j ) ) ⟶ H 0 ( C , L ( − d j ) ) V_{j-1}\otimes H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(d_{j-1}-d_j))\longrightarrow H^0(C,L(-d_j)) V j − 1 ⊗ H 0 ( P 1 , O P 1 ( d j − 1 − d j )) ⟶ H 0 ( C , L ( − d j )) is contained in V j V_j V j 1 ≤ i < j ≤ s 1\leq i<j\leq s 1 ≤ i < j ≤ s Im j ( V i ) ⊆ V j \operatorname{Im}_j(V_i)\subseteq V_j Im j ( V i ) ⊆ V j
V i ⊗ H 0 ( P 1 , O P 1 ( d i − d j ) ) ⟶ H 0 ( C , L ( − d j ) ) . V_i\otimes H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(d_i-d_j))\longrightarrow H^0(C,L(-d_j)). V i ⊗ H 0 ( P 1 , O P 1 ( d i − d j )) ⟶ H 0 ( C , L ( − d j )) . By convention, we set V 0 = { 0 } V_0=\{0\} V 0 = { 0 }
This definition motivates Proposition 1.6 . If [ π : C → P 1 , p ] ∈ H g , k t o t [\pi:C\to\mathbb{P}^1,p]\in\mathcal{H}^{\mathrm{tot}}_{g,k} [ π : C → P 1 , p ] ∈ H g , k tot i ≤ j i\leq j i ≤ j V i V_i V i a a a p p p a , a + k , … , a + k ( d i − d j ) a,a+k,\ldots,a+k(d_i-d_j) a , a + k , … , a + k ( d i − d j ) p p p Im j ( V i ) ⊆ V j \operatorname{Im}_j(V_i)\subseteq V_j Im j ( V i ) ⊆ V j
Definition 2.4 ([15 Section 6] ). Let [ π : C → P 1 , p ] ∈ H g , k t o t [\pi:C\to\mathbb{P}^1,p]\in\mathcal{H}^{\mathrm{tot}}_{g,k} [ π : C → P 1 , p ] ∈ H g , k tot new vanishing orders at layer j j j are the vanishing orders of V j V_j V j p p p Im j ( V j − 1 ) \operatorname{Im}_j(V_{j-1}) Im j ( V j − 1 ) p p p e ⃗ \vec{e} e non-colliding at p p p if, for every 1 ≤ j ≤ s 1\leq j\leq s 1 ≤ j ≤ s j j j k k k k k k V i V_i V i 1 ≤ i < j 1\leq i<j 1 ≤ i < j
We will denote by G e ⃗ , n c ( π , p ) G^{\vec{e},\mathrm{nc}}(\pi,p) G e , nc ( π , p ) e ⃗ \vec{e} e
Proof 1 of Proposition 2.2 . Let ( L , V ) (L,V) ( L , V ) G d , a ( e ⃗ ) r , ∘ ( C , p ) G^{r,\circ}_{d,a(\vec{e})}(C,p) G d , a ( e ) r , ∘ ( C , p ) V = H 0 ( C , L ) V=H^0(C,L) V = H 0 ( C , L ) V ⊊ H 0 ( C , L ) V\subsetneq H^0(C,L) V ⊊ H 0 ( C , L ) L L L r + 2 r+2 r + 2 0 , 1 , … , k − 1 0,1,\ldots,k-1 0 , 1 , … , k − 1 k k k i i i
h 0 ( C , L ( i k ⋅ p ) ) ≥ i k + r + 2 , h^0(C,L(ik\cdot p))\geq ik+r+2, h 0 ( C , L ( ik ⋅ p )) ≥ ik + r + 2 , since its sections have at least that many distinct vanishing orders at p p p Proposition 1.6 . On the other hand, for i i i
h 0 ( C , L ( i k ⋅ p ) ) = i k + r + 1 , h^0(C,L(ik\cdot p))=ik+r+1, h 0 ( C , L ( ik ⋅ p )) = ik + r + 1 , a contradiction.
Using the notation above and that ( L , V ) (L,V) ( L , V )
G d , a ( e ⃗ ) r , ∘ ( C , p ) ⟶ G ord ( e ⃗ ) , n c ( π : C → P 1 , p ) G^{r,\circ}_{d,a(\vec{e})}(C,p)\longrightarrow G^{\operatorname{ord}(\vec{e}),\mathrm{nc}}(\pi:C\to\mathbb{P}^1,p) G d , a ( e ) r , ∘ ( C , p ) ⟶ G ord ( e ) , nc ( π : C → P 1 , p ) into the space of non-colliding ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) ( L , V ) (L,V) ( L , V ) ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) ( L , V 1 , V 2 , … , V s ) (L,V_1,V_2,\ldots,V_s) ( L , V 1 , V 2 , … , V s ) V j = H 0 ( C , L ( − d j ) ) V_j=H^0(C,L(-d_j)) V j = H 0 ( C , L ( − d j )) 1 ≤ j ≤ s 1\leq j\leq s 1 ≤ j ≤ s
V j − 1 ⊗ H 0 ( P 1 , O P 1 ( d j − 1 − d j ) ) ⟶ V j , V_{j-1}\otimes H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(d_{j-1}-d_j))\longrightarrow V_j, V j − 1 ⊗ H 0 ( P 1 , O P 1 ( d j − 1 − d j )) ⟶ V j , show that ( L , V 1 , … , V s ) (L,V_1,\ldots,V_s) ( L , V 1 , … , V s ) ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) V i V_i V i V j V_j V j Im j ( V j ′ ) \operatorname{Im}_j(V_{j'}) Im j ( V j ′ ) j ′ < j j'<j j ′ < j M j M_j M j k k k ( L , V 1 , … , V s ) (L,V_1,\ldots,V_s) ( L , V 1 , … , V s )
We consider the open subspace of G ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}(\pi,p) G ord ( e ) , nc ( π , p ) e ⃗ \vec{e} e V j = H 0 ( C , L ( − d j ) ) V_j=H^0(C,L(-d_j)) V j = H 0 ( C , L ( − d j )) 1 ≤ j ≤ s 1\leq j\leq s 1 ≤ j ≤ s a ( e ⃗ ) a(\vec{e}) a ( e ) inv ( e ⃗ ) \operatorname{inv}(\vec{e}) inv ( e ) d j d_j d j j = 1 j=1 j = 1
The condition that a ( e ⃗ ) ≥ d 1 k a(\vec{e})_{\geq d_1k} a ( e ) ≥ d 1 k ( L , V ) (L,V) ( L , V ) p p p M 1 M_1 M 1 V 1 = H 0 ( C , L ( − d 1 ) ) V_1=H^0(C,L(-d_1)) V 1 = H 0 ( C , L ( − d 1 )) p p p
h 0 ( C , L ( − d 1 k ⋅ p − m 1 , i ⋅ p ) ) ≥ 1 + k 1 − i for each 1 ≤ i ≤ k 1 h^0(C,L(-d_1k\cdot p-m_{1,i}\cdot p))\geq1+k_1-i\quad\text{for each }1\leq i\leq k_1 h 0 ( C , L ( − d 1 k ⋅ p − m 1 , i ⋅ p )) ≥ 1 + k 1 − i for each 1 ≤ i ≤ k 1 ⟺ rank ( V 1 → H 0 ( C , L ( − d 1 k ⋅ p ) ∣ m 1 , i ⋅ p ) ) ≤ i − 1 for each 1 ≤ i ≤ k 1 . \Longleftrightarrow\operatorname{rank}(V_1\to H^0(C,L(-d_1k\cdot p)|_{m_{1,i}\cdot p}))\leq i-1\quad\text{for each }1\leq i\leq k_1. ⟺ rank ( V 1 → H 0 ( C , L ( − d 1 k ⋅ p ) ∣ m 1 , i ⋅ p )) ≤ i − 1 for each 1 ≤ i ≤ k 1 . The locus of ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) L \mathcal{L} L C × G ord ( e ⃗ ) , nc ( π , p ) C\times G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}(\pi,p) C × G ord ( e ) , nc ( π , p ) P = { p } × G ord ( e ⃗ ) , nc ( π , p ) P=\{p\}\times G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}(\pi,p) P = { p } × G ord ( e ) , nc ( π , p ) 1 ≤ j ≤ s 1\leq j\leq s 1 ≤ j ≤ s 1 ≤ i ≤ k j 1\leq i\leq k_j 1 ≤ i ≤ k j
0 ⟶ L ( − ( d j k + m j , i ) ⋅ P ) ⟶ L ( − d j k ⋅ P ) ⟶ L ( − d j k ⋅ P ) / L ( − ( d j k + m j , i ) ⋅ P ) ⟶ 0 0\longrightarrow\mathcal{L}(-(d_jk+m_{j,i})\cdot P)\longrightarrow\mathcal{L}(-d_jk\cdot P)\longrightarrow\mathcal{L}(-d_jk\cdot P)/\mathcal{L}(-(d_jk+m_{j,i})\cdot P)\longrightarrow0 0 ⟶ L ( − ( d j k + m j , i ) ⋅ P ) ⟶ L ( − d j k ⋅ P ) ⟶ L ( − d j k ⋅ P ) / L ( − ( d j k + m j , i ) ⋅ P ) ⟶ 0 and take the rank m j , i m_{j,i} m j , i
F j , i : = π 2 ∗ ( L ( − d j k ⋅ P ) / L ( − ( d j k + m j , i ) ⋅ P ) ) , \mathcal{F}_{j,i}:=\pi_{2*}\bigl(\mathcal{L}(-d_jk\cdot P)/\mathcal{L}(-(d_jk+m_{j,i})\cdot P)\bigr), F j , i := π 2 ∗ ( L ( − d j k ⋅ P ) / L ( − ( d j k + m j , i ) ⋅ P ) ) , defined as a pushforward to G ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}(\pi,p) G ord ( e ) , nc ( π , p )
Let V 1 \mathcal{V}_1 V 1 G ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}(\pi,p) G ord ( e ) , nc ( π , p ) ( L , V 1 , … , V s ) (L,V_1,\ldots,V_s) ( L , V 1 , … , V s ) V 1 V_1 V 1
V 1 ⟶ F 1 , k 1 ↠ F 1 , k 1 − 1 ↠ ⋯ ↠ F 1 , 1 \mathcal{V}_1\longrightarrow\mathcal{F}_{1,k_1}\twoheadrightarrow\mathcal{F}_{1,k_1-1}\twoheadrightarrow\cdots\twoheadrightarrow\mathcal{F}_{1,1} V 1 ⟶ F 1 , k 1 ↠ F 1 , k 1 − 1 ↠ ⋯ ↠ F 1 , 1 and the condition that the entries of a ( e ⃗ ) ≥ d 1 k a(\vec{e})_{\geq d_1k} a ( e ) ≥ d 1 k
rank ( V 1 → F 1 , i ) ≤ i − 1 for each 1 ≤ i ≤ k 1 . \operatorname{rank}(\mathcal{V}_1\to\mathcal{F}_{1,i})\leq i-1\quad\text{for each }1\leq i\leq k_1. rank ( V 1 → F 1 , i ) ≤ i − 1 for each 1 ≤ i ≤ k 1 . The sublocus of G ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}(\pi,p) G ord ( e ) , nc ( π , p )
∑ i = 1 k 1 ( m 1 , i − i + 1 ) = # { ( m , n ) ∈ M 1 × ( M 2 ∪ ⋯ ∪ M s ) ∣ n < m } . \sum_{i=1}^{k_1}(m_{1,i}-i+1)=\#\{(m,n)\in M_1\times(M_2\cup\cdots\cup M_s)\mid n<m\}. i = 1 ∑ k 1 ( m 1 , i − i + 1 ) = # {( m , n ) ∈ M 1 × ( M 2 ∪ ⋯ ∪ M s ) ∣ n < m } . We take G 1 ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_1(\pi,p) G 1 ord ( e ) , nc ( π , p ) H 0 ( C , L ) H^0(C,L) H 0 ( C , L ) a ( e ⃗ ) ≥ d 1 k a(\vec{e})_{\geq d_1k} a ( e ) ≥ d 1 k j j j G j − 1 ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_{j-1}(\pi,p) G j − 1 ord ( e ) , nc ( π , p ) H 0 ( C , L ) H^0(C,L) H 0 ( C , L ) p p p d j − 1 k d_{j-1}k d j − 1 k a ( e ⃗ ) ≥ d j − 1 k a(\vec{e})_{\geq d_{j-1}k} a ( e ) ≥ d j − 1 k d j k d_jk d j k a ( e ⃗ ) ≥ d j k a(\vec{e})_{\geq d_jk} a ( e ) ≥ d j k
In particular, for all ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) ( L , V 1 , V 2 , … , V s ) (L,V_1,V_2,\ldots,V_s) ( L , V 1 , V 2 , … , V s ) V j V_j V j Im j ( V j − 1 ) \operatorname{Im}_{j}(V_{j-1}) Im j ( V j − 1 ) M j M_j M j G j ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_j(\pi,p) G j ord ( e ) , nc ( π , p )
dim V j ( − m j , i p ) / Im j ( V j − 1 ) ( − m j , i p ) ≥ k j + 1 − i for all 1 ≤ i ≤ k j . \dim V_j(-m_{j,i}p)/\operatorname{Im}_{j}(V_{j-1})(-m_{j,i}p)\geq k_j+1-i\quad\text{for all }1\leq i\leq k_j. dim V j ( − m j , i p ) / Im j ( V j − 1 ) ( − m j , i p ) ≥ k j + 1 − i for all 1 ≤ i ≤ k j . We look at the morphisms
0 → Im j ( V j − 1 ) ( − m j , i p ) → Im j ( V j − 1 ) → H 0 ( C , L ( − d j k ⋅ p ) ∣ m j , i ⋅ p ) ↓ ↓ ∥ 0 → H 0 ( C , L ( − ( d j k + m j , i ) ⋅ p ) ) → V j → φ j , i H 0 ( C , L ( − d j k ⋅ p ) ∣ m j , i ⋅ p ) \begin{CD}
0 @>>> \operatorname{Im}_{j}(V_{j-1})(-m_{j,i}p) @>>> \operatorname{Im}_{j}(V_{j-1}) @>>> H^0(C,L(-d_jk\cdot p)|_{m_{j,i}\cdot p}) \\
@. @VVV @VVV @| \\
0 @>>> H^0(C,L(-(d_jk+m_{j,i})\cdot p)) @>>> V_j @>{\varphi_{j,i}}>> H^0(C,L(-d_jk\cdot p)|_{m_{j,i}\cdot p})
\end{CD} 0 0 Im j ( V j − 1 ) ( − m j , i p ) ↓ ⏐ H 0 ( C , L ( − ( d j k + m j , i ) ⋅ p )) Im j ( V j − 1 ) ↓ ⏐ V j φ j , i H 0 ( C , L ( − d j k ⋅ p ) ∣ m j , i ⋅ p ) H 0 ( C , L ( − d j k ⋅ p ) ∣ m j , i ⋅ p ) The conditions
dim V j ( − m j , i p ) / Im j ( V j − 1 ) ( − m j , i p ) ≥ k j + 1 − i for all 1 ≤ i ≤ k j \dim V_j(-m_{j,i}p)/\operatorname{Im}_{j}(V_{j-1})(-m_{j,i}p)\geq k_j+1-i\quad\text{for all }1\leq i\leq k_j dim V j ( − m j , i p ) / Im j ( V j − 1 ) ( − m j , i p ) ≥ k j + 1 − i for all 1 ≤ i ≤ k j above are equivalent, using Snake Lemma, to the maps
V j / I j ( V j − 1 ) ⟶ H 0 ( C , L ( − d j k ⋅ p ) ∣ m j , i ⋅ p ) / φ j , i ( I m ( V j − 1 ) ) \mathcal{V}_j/\mathcal{I}_j(\mathcal{V}_{j-1}) \longrightarrow H^0(C,L(-d_{j k}\cdot p)|_{m_{j,i}\cdot p})/\varphi_{j,i}(\mathcal{I}_m(\mathcal{V}_{j-1})) V j / I j ( V j − 1 ) ⟶ H 0 ( C , L ( − d j k ⋅ p ) ∣ m j , i ⋅ p ) / φ j , i ( I m ( V j − 1 )) having rank less or equal to i − 1 i-1 i − 1
Letting V j \mathcal{V}_j V j G j − 1 ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_{j-1}(\pi,p) G j − 1 ord ( e ) , nc ( π , p ) ( L , V 1 , … , V s ) (L,V_1,\ldots,V_s) ( L , V 1 , … , V s ) V j V_j V j
V j / I j ( V j − 1 ) → φ ‾ j , i F j , i / φ j , i ( I m ( V j − 1 ) ) \mathcal{V}_j/\mathcal{I}_j(\mathcal{V}_{j-1}) \xrightarrow{\overline{\varphi}_{j,i}} \mathcal{F}_{j,i}/\varphi_{j,i}(\mathcal{I}_m(\mathcal{V}_{j-1})) V j / I j ( V j − 1 ) φ j , i F j , i / φ j , i ( I m ( V j − 1 )) of vector bundles on G j − 1 ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_{j-1}(\pi,p) G j − 1 ord ( e ) , nc ( π , p )
The two vector bundles have ranks k j k_j k j m j , i − # { m ∈ M 1 ∪ M 2 ∪ ⋯ ∪ M j − 1 ∣ m < m j , i } m_{j,i}-\#\{m\in M_1\cup M_2\cup\cdots\cup M_{j-1}\mid m<m_{j,i}\} m j , i − # { m ∈ M 1 ∪ M 2 ∪ ⋯ ∪ M j − 1 ∣ m < m j , i }
These morphisms fit into a flag
V j / I j ( V j − 1 ) → φ ‾ j , k j F j , k j / φ j , k j ( I m ( V j − 1 ) ) ↠ ⋯ ↠ F j , 1 / φ j , 1 ( I m ( V j − 1 ) ) . \mathcal{V}_j/\mathcal{I}_j(\mathcal{V}_{j-1}) \xrightarrow{\overline{\varphi}_{j,k_j}} \mathcal{F}_{j,k_j}/\varphi_{j,k_j}(\mathcal{I}_m(\mathcal{V}_{j-1})) \twoheadrightarrow\cdots\twoheadrightarrow\mathcal{F}_{j,1}/\varphi_{j,1}(\mathcal{I}_m(\mathcal{V}_{j-1})). V j / I j ( V j − 1 ) φ j , k j F j , k j / φ j , k j ( I m ( V j − 1 )) ↠ ⋯ ↠ F j , 1 / φ j , 1 ( I m ( V j − 1 )) . In particular, the sublocus of G j − 1 ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_{j-1}(\pi,p) G j − 1 ord ( e ) , nc ( π , p ) H 0 ( C , L ) H^0(C,L) H 0 ( C , L ) p p p a ( e ⃗ ) ≥ d j k a(\vec{e})_{\geq d_{j k}} a ( e ) ≥ d j k
∑ i = 1 k j ( m j , i − # { m ∈ M 1 ∪ M 2 ∪ ⋯ ∪ M j − 1 ∣ m < m j , i } − i + 1 ) . \sum_{i=1}^{k_j}\left(m_{j,i}-\#\{m\in M_1\cup M_2\cup\cdots\cup M_{j-1}\mid m<m_{j,i}\}-i+1\right). i = 1 ∑ k j ( m j , i − # { m ∈ M 1 ∪ M 2 ∪ ⋯ ∪ M j − 1 ∣ m < m j , i } − i + 1 ) . This value is equal to
inv ( M j ) = # { ( m , n ) ∈ M j × ( M j + 1 ∪ ⋯ ∪ M s ) ∣ n < m } . \operatorname{inv}(M_j)=\#\{(m,n)\in M_j\times(M_{j+1}\cup\cdots\cup M_s)\mid n<m\}. inv ( M j ) = # {( m , n ) ∈ M j × ( M j + 1 ∪ ⋯ ∪ M s ) ∣ n < m } . Consequently, the sublocus G j ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_j(\pi,p) G j ord ( e ) , nc ( π , p ) G j − 1 ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_{j-1}(\pi,p) G j − 1 ord ( e ) , nc ( π , p )
⇒ codim ( G j ord ( e ⃗ ) , nc ( π , p ) , G j − 1 ord ( e ⃗ ) , nc ( π , p ) ) ≤ inv ( M j ) . \Rightarrow\operatorname{codim}\left(G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_j(\pi,p),G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_{j-1}(\pi,p)\right)\leq\operatorname{inv}(M_j). ⇒ codim ( G j ord ( e ) , nc ( π , p ) , G j − 1 ord ( e ) , nc ( π , p ) ) ≤ inv ( M j ) . As a consequence
codim ( G s − 1 ord ( e ⃗ ) , nc ( π , p ) , G ord ( e ⃗ ) , nc ( π , p ) ) ≤ ∑ j = 1 s − 1 inv ( M j ) = inv ( e ⃗ ) . \operatorname{codim}\left(G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_{s-1}(\pi,p),G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}(\pi,p)\right)\leq\sum_{j=1}^{s-1}\operatorname{inv}(M_j)=\operatorname{inv}(\vec{e}). codim ( G s − 1 ord ( e ) , nc ( π , p ) , G ord ( e ) , nc ( π , p ) ) ≤ j = 1 ∑ s − 1 inv ( M j ) = inv ( e ) . But G s − 1 ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}_{s-1}(\pi,p) G s − 1 ord ( e ) , nc ( π , p ) G d , a ( e ⃗ ) r , ∘ ( C , p ) G^{r,\circ}_{d,a(\vec{e})}(C,p) G d , a ( e ) r , ∘ ( C , p ) G ord ( e ⃗ ) , nc ( π , p ) G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}(\pi,p) G ord ( e ) , nc ( π , p ) dim ( G ord ( e ⃗ ) , nc ( π , p ) ) ≥ g − u ( e ⃗ ) \dim(G^{\operatorname{ord}(\vec{e}),\operatorname{nc}}(\pi,p))\geq g-u(\vec{e}) dim ( G ord ( e ) , nc ( π , p )) ≥ g − u ( e ) [15 Theorem 6.5] . □
In fact, Proposition 2.2 can be proven independently of the results in [15 ] . As a consequence, our methods give a new proof of the main result of [15 ] about the dimension of Hurwitz–Brill–Noether loci.
Proof 2 of Proposition 2.2 . Let B ( e ⃗ ) B(\vec{e}) B ( e )
B ( e ⃗ ) : = { k e 0 , k e 1 + 1 , … , k e k − 1 + k − 1 } , B(\vec{e}):=\{ke_0,ke_1+1,\ldots,ke_{k-1}+k-1\}, B ( e ) := { k e 0 , k e 1 + 1 , … , k e k − 1 + k − 1 } , i.e. the southernmost entry in each column, in view of Remark 1.17 . We order its elements in increasing order
B ( e ⃗ ) = { b 0 < b 1 < ⋯ < b k − 1 } = { k e i 0 + i 0 < k e i 1 + i 1 < ⋯ < k e i k − 1 + i k − 1 } . B(\vec{e})=\{b_0<b_1<\cdots<b_{k-1}\}=\{ke_{i_0}+i_0<ke_{i_1}+i_1<\cdots<ke_{i_{k-1}}+i_{k-1}\}. B ( e ) = { b 0 < b 1 < ⋯ < b k − 1 } = { k e i 0 + i 0 < k e i 1 + i 1 < ⋯ < k e i k − 1 + i k − 1 } . and also denote by a ( e ⃗ ) > b i a(\vec{e})_{>b_i} a ( e ) > b i a ( e ⃗ ) a(\vec{e}) a ( e ) b i b_i b i c i = min a ( e ⃗ ) > b i c_i=\min a(\vec{e})_{>b_i} c i = min a ( e ) > b i 0 ≤ i ≤ k − 2 0\leq i\leq k-2 0 ≤ i ≤ k − 2
Our goal is to realize the pointed Hurwitz–Brill–Noether locus W e ⃗ ( π , p ) = W g + r , a ( e ⃗ ) r , ∘ ( C , p ) W^{\vec{e}}(\pi,p)=W^{r,\circ}_{g+r,a(\vec{e})}(C,p) W e ( π , p ) = W g + r , a ( e ) r , ∘ ( C , p )
h 0 ( C , L − ( e i ⋅ k + i ) ⋅ p ) ≥ 1 + ∑ e j ≥ e i ( e j − e i ) + # { j ∣ j > i and e j ≥ e i } h^0(C,L-(e_i\cdot k+i)\cdot p)\geq1+\sum_{e_j\geq e_i}(e_j-e_i)+\#\{j\mid j>i\text{ and }e_j\geq e_i\} h 0 ( C , L − ( e i ⋅ k + i ) ⋅ p ) ≥ 1 + e j ≥ e i ∑ ( e j − e i ) + # { j ∣ j > i and e j ≥ e i } for each $0\leq i\leq k-1 in an inductive manner, and emphasize that all other vanishing conditions are automatically satisfied because of Proposition 1.6.
Figure 1 (PDF p. 19).
This is an example for k = 4 k=4 k = 4 e ⃗ = ( 2 , 4 , 1 , 0 ) \vec{e}=(2,4,1,0) e = ( 2 , 4 , 1 , 0 ) B ( e ⃗ ) = { 3 , 6 , 8 , 17 } B(\vec{e})=\{3,6,8,17\} B ( e ) = { 3 , 6 , 8 , 17 } i = 0 i=0 i = 0
∑ e j ≥ e i ( e j − e i ) = 2 \sum_{e_j\geq e_i}(e_j-e_i)=2 e j ≥ e i ∑ ( e j − e i ) = 2 counts the vanishing orders on higher rows than 8 = 4 ⋅ e 0 + 0 8=4\cdot e_0+0 8 = 4 ⋅ e 0 + 0
∑ e j ≥ e i ( e j − e i ) + # { j ∣ j > i and e j ≥ e i } = 3 \sum_{e_j\geq e_i}(e_j-e_i)+\#\{j\mid j>i\text{ and }e_j\geq e_i\}=3 e j ≥ e i ∑ ( e j − e i ) + # { j ∣ j > i and e j ≥ e i } = 3 counts the vanishing orders strictly greater than 8 8 8
h 0 ( C , L − 8 p ) ≥ 1 + 2 + 1 = 4. h^0(C,L-8p)\geq1+2+1=4. h 0 ( C , L − 8 p ) ≥ 1 + 2 + 1 = 4. We start with the condition
h 0 ( C , L ( − b k − 1 ⋅ p ) ) ≥ 1 h^0(C,L(-b_{k-1}\cdot p))\geq1 h 0 ( C , L ( − b k − 1 ⋅ p )) ≥ 1 which we want to realise as a degeneracy locus on Pic g + r ( C ) \operatorname{Pic}^{g+r}(C) Pic g + r ( C )
Let m > 0 m>0 m > 0 L ⊗ O C ( m ⋅ p ) L\otimes\mathcal{O}_C(m\cdot p) L ⊗ O C ( m ⋅ p ) L ∈ Pic g + r ( C ) L\in\operatorname{Pic}^{g+r}(C) L ∈ Pic g + r ( C )
0 ⟶ L ⊗ O C ( − b k − 1 ⋅ p ) ⟶ L ⊗ O C ( m ⋅ p ) ⟶ L ⊗ O C ( m ⋅ p ) ∣ ( b k − 1 + m ) ⋅ p ⟶ 0. 0\longrightarrow L\otimes\mathcal{O}_C(-b_{k-1}\cdot p)\longrightarrow L\otimes\mathcal{O}_C(m\cdot p)\longrightarrow L\otimes\mathcal{O}_C(m\cdot p)|_{(b_{k-1}+m)\cdot p}\longrightarrow0. 0 ⟶ L ⊗ O C ( − b k − 1 ⋅ p ) ⟶ L ⊗ O C ( m ⋅ p ) ⟶ L ⊗ O C ( m ⋅ p ) ∣ ( b k − 1 + m ) ⋅ p ⟶ 0. Looking at the exact sequence obtained by taking global sections, the condition h 0 ( C , L − b k − 1 ⋅ p ) ≥ 1 h^0(C,L-b_{k-1}\cdot p)\geq1 h 0 ( C , L − b k − 1 ⋅ p ) ≥ 1
H 0 ( L ⊗ O C ( m ⋅ p ) ) ⟶ H 0 ( C , L ⊗ O C ( m ⋅ p ) ∣ ( b k − 1 + m ) ⋅ p ) H^0(L\otimes\mathcal{O}_C(m\cdot p))\longrightarrow H^0(C,L\otimes\mathcal{O}_C(m\cdot p)|_{(b_{k-1}+m)\cdot p}) H 0 ( L ⊗ O C ( m ⋅ p )) ⟶ H 0 ( C , L ⊗ O C ( m ⋅ p ) ∣ ( b k − 1 + m ) ⋅ p ) having rank at most m + r m+r m + r
Consider the product space C × Pic g + r ( C ) C\times\operatorname{Pic}^{g+r}(C) C × Pic g + r ( C ) π 1 \pi_1 π 1 π 2 \pi_2 π 2 L \mathcal{L} L C × Pic g + r ( C ) C\times\operatorname{Pic}^{g+r}(C) C × Pic g + r ( C ) D : = m ⋅ { p } × Pic g + r ( C ) D:=m\cdot\{p\}\times\operatorname{Pic}^{g+r}(C) D := m ⋅ { p } × Pic g + r ( C )
P i : = ( b i + m ) ⋅ { p } × Pic g + r ( C ) P_i:=(b_i+m)\cdot\{p\}\times\operatorname{Pic}^{g+r}(C) P i := ( b i + m ) ⋅ { p } × Pic g + r ( C ) for each $0\leq i\leq k-1.
The locus in Pic g + r ( C ) \operatorname{Pic}^{g+r}(C) Pic g + r ( C ) h 0 ( C , L − b k − 1 ⋅ p ) ≥ 1 h^0(C,L-b_{k-1}\cdot p)\geq1 h 0 ( C , L − b k − 1 ⋅ p ) ≥ 1
E : = π 2 ∗ L ( D ) ⟶ π 2 ∗ ( L ( D ) ⊗ O P k − 1 ) \mathcal{E}:=\pi_{2*}\mathcal{L}(D)\longrightarrow\pi_{2*}(\mathcal{L}(D)\otimes\mathcal{O}_{P_{k-1}}) E := π 2 ∗ L ( D ) ⟶ π 2 ∗ ( L ( D ) ⊗ O P k − 1 ) has rank at most m + r m+r m + r r + 1 + m r+1+m r + 1 + m b k − 1 + m b_{k-1}+m b k − 1 + m ( b k − 1 − r ) (b_{k-1}-r) ( b k − 1 − r ) Pic g + r ( C ) \operatorname{Pic}^{g+r}(C) Pic g + r ( C )
We denote by X k − 1 X_{k-1} X k − 1 L L L h 0 ( C , L − b k − 1 ⋅ p ) = 1 h^0(C,L-b_{k-1}\cdot p)=1 h 0 ( C , L − b k − 1 ⋅ p ) = 1 h 0 ( C , L − ( b k − 1 + 1 ) ⋅ p ) = 0 h^0(C,L-(b_{k-1}+1)\cdot p)=0 h 0 ( C , L − ( b k − 1 + 1 ) ⋅ p ) = 0 X k − 1 X_{k-1} X k − 1 ( b k − 1 − r ) (b_{k-1}-r) ( b k − 1 − r ) Pic g + r ( C ) \operatorname{Pic}^{g+r}(C) Pic g + r ( C )
Using Proposition 1.6, it follows that all elements of a ( e ⃗ ) > b k − 2 a(\vec{e})_{>b_{k-2}} a ( e ) > b k − 2 L L L X k − 1 X_{k-1} X k − 1 i k − 2 i_{k-2} i k − 2 b k − 2 = k e i k − 2 + i k − 2 b_{k-2}=ke_{i_{k-2}}+i_{k-2} b k − 2 = k e i k − 2 + i k − 2
h 0 ( C , L − c k − 2 ⋅ p ) ≥ ∑ j ≥ k − 2 ( e i j − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } h^0(C,L-c_{k-2}\cdot p)\geq\sum_{j\geq k-2}(e_{i_j}-e_{i_{k-2}})+\#\{j\mid j>i_{k-2}\text{ and }e_j\geq e_{i_{k-2}}\} h 0 ( C , L − c k − 2 ⋅ p ) ≥ j ≥ k − 2 ∑ ( e i j − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } and we denote by Y k − 1 Y_{k-1} Y k − 1 X k − 1 X_{k-1} X k − 1 L L L c k − 2 c_{k-2} c k − 2 a ( e ⃗ ) > b k − 2 a(\vec{e})>b_{k-2} a ( e ) > b k − 2
We look at the sublocus of Y k − 1 Y_{k-1} Y k − 1
h 0 ( C , L − b k − 2 ⋅ p ) ≥ 1 + ∑ j ≥ k − 2 ( e i j − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } h^0(C,L-b_{k-2}\cdot p)\geq1+\sum_{j\geq k-2}(e_{ij}-e_{i_{k-2}})+\#\{j\mid j>i_{k-2}\text{ and }e_j\geq e_{i_{k-2}}\} h 0 ( C , L − b k − 2 ⋅ p ) ≥ 1 + j ≥ k − 2 ∑ ( e ij − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } and we want to estimate its codimension.
Let
Q i : = ( c i + m ) ⋅ { p } × Pic g + r ( C ) Q_i:=(c_i+m)\cdot\{p\}\times\operatorname{Pic}^{g+r}(C) Q i := ( c i + m ) ⋅ { p } × Pic g + r ( C ) for each 0 ≤ i ≤ k − 2 0\leq i\leq k-2 0 ≤ i ≤ k − 2
E ⟶ π 2 ∗ ( L ( D ) ⊗ O Q k − 2 ) ⟶ π 2 ∗ ( L ( D ) ⊗ O P k − 2 ) \mathcal{E}\longrightarrow\pi_{2*}(\mathcal{L}(D)\otimes\mathcal{O}_{Q_{k-2}})\longrightarrow\pi_{2*}(\mathcal{L}(D)\otimes\mathcal{O}_{P_{k-2}}) E ⟶ π 2 ∗ ( L ( D ) ⊗ O Q k − 2 ) ⟶ π 2 ∗ ( L ( D ) ⊗ O P k − 2 ) which we restrict to Y k − 1 Y_{k-1} Y k − 1 Y k − 1 Y_{k-1} Y k − 1 L L L
r + m − ∑ j ≥ k − 2 ( e i j − e i k − 2 ) − # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } . r+m-\sum_{j\geq k-2}(e_{ij}-e_{i_{k-2}})-\#\{j\mid j>i_{k-2}\text{ and }e_j\geq e_{i_{k-2}}\}. r + m − j ≥ k − 2 ∑ ( e ij − e i k − 2 ) − # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } . Because we have
h 0 ( C , L − c k − 2 ⋅ p ) = ∑ j ≥ k − 2 ( e i j − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } h^0(C,L-c_{k-2}\cdot p)=\sum_{j\geq k-2}(e_{ij}-e_{i_{k-2}})+\#\{j\mid j>i_{k-2}\text{ and }e_j\geq e_{i_{k-2}}\} h 0 ( C , L − c k − 2 ⋅ p ) = j ≥ k − 2 ∑ ( e ij − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } on Y k − 1 Y_{k-1} Y k − 1
φ k − 2 : E ⟶ π 2 ∗ ( L ( D ) ⊗ O Q k − 2 ) \varphi_{k-2}:\mathcal{E}\longrightarrow\pi_{2*}(\mathcal{L}(D)\otimes\mathcal{O}_{Q_{k-2}}) φ k − 2 : E ⟶ π 2 ∗ ( L ( D ) ⊗ O Q k − 2 ) has rank r + 1 + m − ∑ j ≥ k − 2 ( e i j − e i k − 2 ) − # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } r+1+m-\sum_{j\geq k-2}(e_{ij}-e_{i_{k-2}})-\#\{j\mid j>i_{k-2}\text{ and }e_j\geq e_{i_{k-2}}\} r + 1 + m − ∑ j ≥ k − 2 ( e ij − e i k − 2 ) − # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } L L L Y k − 1 Y_{k-1} Y k − 1 Im ( φ k − 2 ) \operatorname{Im}(\varphi_{k-2}) Im ( φ k − 2 ) Y k − 1 Y_{k-1} Y k − 1
r + 1 + m − ∑ j ≥ k − 2 ( e i j − e i k − 2 ) − # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } . r+1+m-\sum_{j\geq k-2}(e_{ij}-e_{i_{k-2}})-\#\{j\mid j>i_{k-2}\text{ and }e_j\geq e_{i_{k-2}}\}. r + 1 + m − j ≥ k − 2 ∑ ( e ij − e i k − 2 ) − # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } . We can view our required sublocus of Y k − 1 Y_{k-1} Y k − 1
Im ( φ k − 2 ) ⟶ π 2 ∗ ( L ( D ) ⊗ O P k − 2 ) \operatorname{Im}(\varphi_{k-2})\longrightarrow\pi_{2*}(\mathcal{L}(D)\otimes\mathcal{O}_{P_{k-2}}) Im ( φ k − 2 ) ⟶ π 2 ∗ ( L ( D ) ⊗ O P k − 2 ) has rank at most
r + m − ∑ j ≥ k − 2 ( e i j − e i k − 2 ) − # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } . r+m-\sum_{j\geq k-2}(e_{ij}-e_{i_{k-2}})-\#\{j\mid j>i_{k-2}\text{ and }e_j\geq e_{i_{k-2}}\}. r + m − j ≥ k − 2 ∑ ( e ij − e i k − 2 ) − # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } . The vector bundles Im ( φ k − 2 ) \operatorname{Im}(\varphi_{k-2}) Im ( φ k − 2 ) π 2 ∗ ( L ( D ) ⊗ O P k − 2 ) \pi_{2*}(\mathcal{L}(D)\otimes\mathcal{O}_{P_{k-2}}) π 2 ∗ ( L ( D ) ⊗ O P k − 2 )
r + m + 1 − ∑ j ≥ k − 2 ( e i j − e i k − 2 ) − # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } r+m+1-\sum_{j\geq k-2}(e_{ij}-e_{i_{k-2}})-\#\{j\mid j>i_{k-2}\text{ and }e_j\geq e_{i_{k-2}}\} r + m + 1 − j ≥ k − 2 ∑ ( e ij − e i k − 2 ) − # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } and b k − 2 + m b_{k-2}+m b k − 2 + m Y k − 1 Y_{k-1} Y k − 1
b k − 2 − r + ∑ j ≥ k − 2 ( e i j − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } . b_{k-2}-r+\sum_{j\geq k-2}(e_{ij}-e_{i_{k-2}})+\#\{j\mid j>i_{k-2}\text{ and }e_j\geq e_{i_{k-2}}\}. b k − 2 − r + j ≥ k − 2 ∑ ( e ij − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } . We denote by X k − 2 X_{k-2} X k − 2
h 0 ( L − b k − 2 ⋅ p ) = 1 + ∑ j ≥ k − 2 ( e i j − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } h^0(L-b_{k-2}\cdot p)=1+\sum_{j\geq k-2}(e_{ij}-e_{i_{k-2}})+\#\{j\mid j>i_{k-2}\text{ and }e_j\geq e_{i_{k-2}}\} h 0 ( L − b k − 2 ⋅ p ) = 1 + j ≥ k − 2 ∑ ( e ij − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } and
h 0 ( L − ( b k − 2 + 1 ) ⋅ p ) = ∑ j ≥ k − 2 ( e i j − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } . h^0(L-(b_{k-2}+1)\cdot p)=\sum_{j\geq k-2}(e_{ij}-e_{i_{k-2}})+\#\{j\mid j>i_{k-2}\text{ and }e_j\geq e_{i_{k-2}}\}. h 0 ( L − ( b k − 2 + 1 ) ⋅ p ) = j ≥ k − 2 ∑ ( e ij − e i k − 2 ) + # { j ∣ j > i k − 2 and e j ≥ e i k − 2 } . We now define Y k − 2 Y_{k-2} Y k − 2 X k − 2 X_{k-2} X k − 2 c k − 3 c_{k-3} c k − 3 a ( e ⃗ ) > b k − 3 \mathbf{a}(\vec{e})>b_{k-3} a ( e ) > b k − 3
Inductively, we obtain a series of inclusions
Y 0 ⊆ Y 1 ⊆ ⋯ ⊆ Y k − 1 ⊆ Pic g + r ( C ) , Y_0 \subseteq Y_1 \subseteq\cdots\subseteq Y_{k-1} \subseteq\operatorname{Pic}^{g+r}(C), Y 0 ⊆ Y 1 ⊆ ⋯ ⊆ Y k − 1 ⊆ Pic g + r ( C ) , with Y t Y_t Y t p p p c t − 1 c_{t-1} c t − 1 a ( e ⃗ ) > b t − 1 \mathbf{a}(\vec{e})>b_{t-1} a ( e ) > b t − 1 Y 0 Y_0 Y 0 Pic g + r ( C ) \operatorname{Pic}^{g+r}(C) Pic g + r ( C ) g g + r r g^{r}_{g+r} g g + r r a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) p p p Y 0 = W e ⃗ ( π , p ) Y_0=W^{\vec{e}}(\pi,p) Y 0 = W e ( π , p )
The codimension estimate that we get using our inductive argument is
codim ( Y t − 1 ; Y t ) ≤ b t − 1 − r + ∑ j ≥ t − 1 ( e i j − e i t − 1 ) + # { j ∣ j > i t − 1 and e j ≥ e i t − 1 } . \operatorname{codim}(Y_{t-1};Y_t) \leq b_{t-1}-r+\sum_{j\geq t-1}(e_{ij}-e_{i_{t-1}})+\#\{j\mid j>i_{t-1}\text{ and }e_j\geq e_{i_{t-1}}\}. codim ( Y t − 1 ; Y t ) ≤ b t − 1 − r + j ≥ t − 1 ∑ ( e ij − e i t − 1 ) + # { j ∣ j > i t − 1 and e j ≥ e i t − 1 } . Adding up all the codimension we get that
codim ( Y 0 ; Pic g + r ( C ) ) ≤ ∑ i = 0 k − 1 b i − k r + ∑ e i ≥ e j ( e i − e j ) + # { ( i , j ) ∣ j > i and e j ≥ e i } . ≤ k ∑ i = 0 k − 1 e i + ∑ i = 0 k − 1 i − k r + ∑ e i ≥ e j ( e i − e j ) + # { ( i , j ) ∣ j > i and e j ≥ e i } ≤ k ( r + 1 − k ) + ∑ i = 0 k − 1 i − k r + ∑ e i ≥ e j ( e i − e j ) + # { ( i , j ) ∣ j > i and e j ≥ e i } ≤ − k ( k − 1 ) 2 + ∑ e i ≥ e j ( e i − e j ) + # { ( i , j ) ∣ j > i and e j ≥ e i } . \begin{aligned}
\operatorname{codim}(Y_0;\operatorname{Pic}^{g+r}(C))\leq\sum_{i=0}^{k-1}b_i-kr+\sum_{e_i\geq e_j}(e_i-e_j)+\#\{(i,j)\mid j>i\text{ and }e_j\geq e_i\}. \\
&\leq k\sum_{i=0}^{k-1}e_i+\sum_{i=0}^{k-1}i-kr+\sum_{e_i\geq e_j}(e_i-e_j)+\#\{(i,j)\mid j>i\text{ and }e_j\geq e_i\} \\
&\leq k(r+1-k)+\sum_{i=0}^{k-1}i-kr+\sum_{e_i\geq e_j}(e_i-e_j)+\#\{(i,j)\mid j>i\text{ and }e_j\geq e_i\} \\
&\leq-\frac{k(k-1)}{2}+\sum_{e_i\geq e_j}(e_i-e_j)+\#\{(i,j)\mid j>i\text{ and }e_j\geq e_i\}.
\end{aligned} codim ( Y 0 ; Pic g + r ( C )) ≤ i = 0 ∑ k − 1 b i − k r + e i ≥ e j ∑ ( e i − e j ) + # {( i , j ) ∣ j > i and e j ≥ e i } . ≤ k i = 0 ∑ k − 1 e i + i = 0 ∑ k − 1 i − k r + e i ≥ e j ∑ ( e i − e j ) + # {( i , j ) ∣ j > i and e j ≥ e i } ≤ k ( r + 1 − k ) + i = 0 ∑ k − 1 i − k r + e i ≥ e j ∑ ( e i − e j ) + # {( i , j ) ∣ j > i and e j ≥ e i } ≤ − 2 k ( k − 1 ) + e i ≥ e j ∑ ( e i − e j ) + # {( i , j ) ∣ j > i and e j ≥ e i } . But we immediately check the equalities
∑ e i ≥ e j ( e i − e j ) − k ( k − 1 ) 2 = u ( e ⃗ ) − # { ( i , j ) ∣ i < j and e i = e j } \sum_{e_i\geq e_j}(e_i-e_j)-\frac{k(k-1)}{2}=u(\vec{e})-\#\{(i,j)\mid i<j\text{ and }e_i=e_j\} e i ≥ e j ∑ ( e i − e j ) − 2 k ( k − 1 ) = u ( e ) − # {( i , j ) ∣ i < j and e i = e j } and
# { ( i , j ) ∣ j > i and e j ≥ e i } = inv ( e ⃗ ) + # { ( i , j ) ∣ i < j and e i = e j } . \#\{(i,j)\mid j>i\text{ and }e_j\geq e_i\}=\operatorname{inv}(\vec{e})+\#\{(i,j)\mid i<j\text{ and }e_i=e_j\}. # {( i , j ) ∣ j > i and e j ≥ e i } = inv ( e ) + # {( i , j ) ∣ i < j and e i = e j } . Putting the two equalities together we obtain that Y 0 Y_0 Y 0 g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec{e})-\operatorname{inv}(\vec{e}) g − u ( e ) − inv ( e )
Putting together Proposition 2.1 and Proposition 2.2 we obtain a pointed version of the main result in [15 ] .
Theorem 2.5. Let [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^{1},p] [ π : C → P 1 , p ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot e ⃗ ∈ Z ≥ 0 k \vec{e}\in\mathbb{Z}^{k}_{\geq0} e ∈ Z ≥ 0 k W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p )
dim W e ⃗ ( π , p ) = g − u ( e ⃗ ) − inv ( e ⃗ ) \dim W^{\vec{e}}(\pi,p)=g-u(\vec{e})-\operatorname{inv}(\vec{e}) dim W e ( π , p ) = g − u ( e ) − inv ( e ) In fact, whenever g − u ( e ⃗ ) − inv ( e ⃗ ) ≥ 0 g-u(\vec{e})-\operatorname{inv}(\vec{e})\geq0 g − u ( e ) − inv ( e ) ≥ 0 Theorem 3.5 .
Remark 2.6. Let [ π , p ] [\pi,p] [ π , p ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot [14 ] whose associated vector has the form
e ⃗ = ( a + 1 , … , a + 1 ⏟ k 1 times , a , … , a ⏟ k 2 times , 1 , … , 1 ⏟ k 3 times , 0 , … , 0 ⏟ k 4 times ) , \vec{e}=(\underbrace{a+1,\ldots,a+1}_{k_1\text{ times}},\underbrace{a,\ldots,a}_{k_2\text{ times}},\underbrace{1,\ldots,1}_{k_3\text{ times}},\underbrace{0,\ldots,0}_{k_4\text{ times}}), e = ( k 1 times a + 1 , … , a + 1 , k 2 times a , … , a , k 3 times 1 , … , 1 , k 4 times 0 , … , 0 ) , where a ≥ 2 a \ge2 a ≥ 2 k i k_i k i k 1 + k 2 + k 3 + k 4 = k k_1+k_2+k_3+k_4=k k 1 + k 2 + k 3 + k 4 = k Proposition 2.2 realizes the maximal Hurwitz–Brill–Noether locus as a pointed Brill–Noether locus of expected dimension.
Proof. Consider the rank degeneracy conditions
h 0 ( C , L − ( a k + k + i ) p ) ≥ k 1 − i for all 0 ≤ i ≤ k 1 − 1 h^0(C,L-(ak+k+i)p)\ge k_1-i \text{ for all } 0\le i\le k_1-1 h 0 ( C , L − ( ak + k + i ) p ) ≥ k 1 − i for all 0 ≤ i ≤ k 1 − 1 and
h 0 ( C , L − ( a k + k 1 + i ) p ) ≥ k 1 + k 2 − i for all 0 ≤ i ≤ k 2 − 1. h^0(C,L-(ak+k_1+i)p)\ge k_1+k_2-i \text{ for all } 0\le i\le k_2-1. h 0 ( C , L − ( ak + k 1 + i ) p ) ≥ k 1 + k 2 − i for all 0 ≤ i ≤ k 2 − 1. According to the second proof of Proposition 2.2 , these are all the non-trivial rank conditions we need to cut out W e ⃗ ( π , p ) W^{\vec e}(\pi,p) W e ( π , p )
Hence, if we denote
a : = ( 0 , 1 , … , r − k 1 − k 2 , a k + k 1 , … , a k + k 1 + k 2 − 1 , a k + k , … , a k + k + k 1 − 1 ) , \mathbf{a}:=(0,1,\ldots,r-k_1-k_2,ak+k_1,\ldots,ak+k_1+k_2-1,ak+k,\ldots,ak+k+k_1-1), a := ( 0 , 1 , … , r − k 1 − k 2 , ak + k 1 , … , ak + k 1 + k 2 − 1 , ak + k , … , ak + k + k 1 − 1 ) , we obtain that W e ⃗ ( π , p ) W^{\vec e}(\pi,p) W e ( π , p ) W g + r , a r ( C , p ) W^{r}_{g+r,\mathbf{a}}(C,p) W g + r , a r ( C , p )
Figure 4. Expected dimension: g − 20 g-20 g − 20
In both cases, the expected codimension is the same. In one case, it is computed as the sum of displacements of southernmost squares, while in the second case it is computed as the sum of all displacement. The terms are the same (in our example: 6 + 6 + 8 6+6+8 6 + 6 + 8
Consider the vector
f ⃗ : = ( e 0 , … , e i − 1 , e i + 1 , e i , … , e k − 1 ) \vec f:=(e_0,\ldots,e_{i-1},e_{i+1},e_i,\ldots,e_{k-1}) f := ( e 0 , … , e i − 1 , e i + 1 , e i , … , e k − 1 ) so that t i + 1 ( f ⃗ ) = e ⃗ t_{i+1}(\vec f)=\vec e t i + 1 ( f ) = e Proposition 2.2 can be adapted to show that the irreducible components of
W e ⃗ ( π , p ) ∪ W f ⃗ ( π , p ) W^{\vec e}(\pi,p)\cup W^{\vec f}(\pi,p) W e ( π , p ) ∪ W f ( π , p ) have dimension at least g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec e)-\operatorname{inv}(\vec e) g − u ( e ) − inv ( e ) Theorem 2.5 implies that
dim W f ⃗ ( π , p ) = g − 1 − u ( e ⃗ ) − inv ( e ⃗ ) , \dim W^{\vec f}(\pi,p)=g-1-u(\vec e)-\operatorname{inv}(\vec e), dim W f ( π , p ) = g − 1 − u ( e ) − inv ( e ) , and hence W f ⃗ ( π , p ) ⊆ W e ⃗ ( π , p ) ‾ W^{\vec f}(\pi,p)\subseteq\overline{W^{\vec e}(\pi,p)} W f ( π , p ) ⊆ W e ( π , p )
Proposition 2.7. Let [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^1,p] [ π : C → P 1 , p ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot 0 ≤ i ≤ k − 2 0\le i\le k-2 0 ≤ i ≤ k − 2 e i > e i + 1 e_i>e_{i+1} e i > e i + 1
f ⃗ : = ( e 0 , … , e i − 1 , e i + 1 , e i , … , e k − 1 ) . \vec f:=(e_0,\ldots,e_{i-1},e_{i+1},e_i,\ldots,e_{k-1}). f := ( e 0 , … , e i − 1 , e i + 1 , e i , … , e k − 1 ) . Then we have the inclusion
W f ⃗ ( π , p ) ⊆ W e ⃗ ( π , p ) ‾ . W^{\vec f}(\pi,p)\subseteq\overline{W^{\vec e}(\pi,p)}. W f ( π , p ) ⊆ W e ( π , p ) . The same holds if e k − 1 + 1 > e 0 e_{k-1}+1>e_0 e k − 1 + 1 > e 0
f ⃗ = ( e k − 1 + 1 , e 1 , e 2 … , e k − 2 , e 0 − 1 ) . \vec{f}=(e_{k-1}+1,e_1,e_2\ldots,e_{k-2},e_0-1). f = ( e k − 1 + 1 , e 1 , e 2 … , e k − 2 , e 0 − 1 ) . Proof. There is nothing to prove if W f ⃗ ( π , p ) W^{\vec{f}}(\pi,p) W f ( π , p ) Theorem 2.5 gives
dim W f ⃗ ( π , p ) = g − u ( e ⃗ ) − inv ( e ⃗ ) − 1. \dim W^{\vec{f}}(\pi,p)=g-u(\vec{e})-\operatorname{inv}(\vec{e})-1. dim W f ( π , p ) = g − u ( e ) − inv ( e ) − 1. It therefore suffices to show that every irreducible component of W e ⃗ ( π , p ) ∪ W f ⃗ ( π , p ) W^{\vec{e}}(\pi,p)\cup W^{\vec{f}}(\pi,p) W e ( π , p ) ∪ W f ( π , p ) g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec{e})-\operatorname{inv}(\vec{e}) g − u ( e ) − inv ( e ) W f ⃗ ( π , p ) W^{\vec{f}}(\pi,p) W f ( π , p ) W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p )
We use the construction in the first proof of Proposition 2.2 . We will realize W e ⃗ ( π , p ) ∪ W f ⃗ ( π , p ) W^{\vec{e}}(\pi,p)\cup W^{\vec{f}}(\pi,p) W e ( π , p ) ∪ W f ( π , p ) ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e )
Let ( j 1 , i 1 ) (j_1,i_1) ( j 1 , i 1 ) ( j 2 , i 2 ) (j_2,i_2) ( j 2 , i 2 )
m j 1 , i 1 = i and m j 2 , i 2 = i + 1 , m_{j_1,i_1}=i \qquad\text{and}\qquad m_{j_2,i_2}=i+1, m j 1 , i 1 = i and m j 2 , i 2 = i + 1 , in particular d j 1 = e i > e i + 1 = d j 2 d_{j_1}=e_i>e_{i+1}=d_{j_2} d j 1 = e i > e i + 1 = d j 2 j 1 < j 2 j_1<j_2 j 1 < j 2
Recall that in the proof of Proposition 2.2 , we imposed the rank conditions
rk ( V j / Im j ( V j − 1 ) ⟶ H 0 ( C , L ( − d j k ⋅ p ) ∣ m j , i ⋅ p ) / φ j , i ( Im j ( V j − 1 ) ) ) ≤ i − 1 \operatorname{rk}\left(V_j/\operatorname{Im}_{j}(V_{j-1})\longrightarrow H^0(C,L(-d_{j_k}\cdot p)|_{m_{j,i}\cdot p})/\varphi_{j,i}(\operatorname{Im}_{j}(V_{j-1}))\right)\leq i-1 rk ( V j / Im j ( V j − 1 ) ⟶ H 0 ( C , L ( − d j k ⋅ p ) ∣ m j , i ⋅ p ) / φ j , i ( Im j ( V j − 1 )) ) ≤ i − 1 for 1 ≤ j ≤ s 1\leq j\leq s 1 ≤ j ≤ s 1 ≤ i ≤ k j 1\leq i\leq k_j 1 ≤ i ≤ k j j j j m j , i m_{j,i} m j , i
h 0 ( C , L ( − ( d j k + m j , i ) ⋅ p ) ) = 1 + h 0 ( C , L ( − ( d j k + m j , i + 1 ) ⋅ p ) ) . h^0(C,L(-(d_{j_k}+m_{j,i})\cdot p))=1+h^0(C,L(-(d_{j_k}+m_{j,i}+1)\cdot p)). h 0 ( C , L ( − ( d j k + m j , i ) ⋅ p )) = 1 + h 0 ( C , L ( − ( d j k + m j , i + 1 ) ⋅ p )) . The imposed vanishing conditions ensure that φ j , i ( Im j ( V j − 1 ) ) \varphi_{j,i}(\operatorname{Im}_{j}(V_{j-1})) φ j , i ( Im j ( V j − 1 ))
F j , i / φ j , i ( Im j ( V j − 1 ) ) \mathcal{F}_{j,i}/\varphi_{j,i}(\operatorname{Im}_{j}(V_{j-1})) F j , i / φ j , i ( Im j ( V j − 1 )) is a vector bundle of rank
m j , i − # { m ∈ M 1 ∪ ⋯ ∪ M j − 1 ∣ m < m j , i } . m_{j,i}-\#\{m\in M_1\cup\cdots\cup M_{j-1}\mid m<m_{j,i}\}. m j , i − # { m ∈ M 1 ∪ ⋯ ∪ M j − 1 ∣ m < m j , i } . Finally, the flag degeneracy conditions, together with the vanishing order conditions cut out exactly the locus W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p )
For our flag degeneracy conditions to cut out W e ⃗ ( π , p ) ∪ W f ⃗ ( π , p ) W^{\vec{e}}(\pi,p)\cup W^{\vec{f}}(\pi,p) W e ( π , p ) ∪ W f ( π , p )
The flag degeneracy condition on layer j 2 j_2 j 2 Proposition 2.2 , but we relax the vanishing order condition: we restrict to the open locus on which either
d j 1 k + m j 1 , i 1 or d j 1 k + m j 1 , i 1 + 1 d_{j_1}k+m_{j_1,i_1}\qquad\text{or}\qquad d_{j_1}k+m_{j_1,i_1}+1 d j 1 k + m j 1 , i 1 or d j 1 k + m j 1 , i 1 + 1 is a vanishing order; or equivalently
h 0 ( C , L ( − ( d j 1 k + m j 1 , i 1 ) ⋅ p ) ) = 1 + h 0 ( C , L ( − ( d j 1 k + m j 1 , i 1 + 2 ) ⋅ p ) ) . h^0(C,L(-(d_{j_1}k+m_{j_1,i_1})\cdot p))=1+h^0(C,L(-(d_{j_1}k+m_{j_1,i_1}+2)\cdot p)). h 0 ( C , L ( − ( d j 1 k + m j 1 , i 1 ) ⋅ p )) = 1 + h 0 ( C , L ( − ( d j 1 k + m j 1 , i 1 + 2 ) ⋅ p )) . Remark: In the Proof of Proposition 2.2 , we imposed the condition
h 0 ( C , L ( − ( d j 1 k + m j 1 , i 1 ) ⋅ p ) ) = 1 + h 0 ( C , L ( − ( d j 1 k + m j 1 , i 1 + 1 ) ⋅ p ) ) , h^0(C,L(-(d_{j_1}k+m_{j_1,i_1})\cdot p))=1+h^0(C,L(-(d_{j_1}k+m_{j_1,i_1}+1)\cdot p)), h 0 ( C , L ( − ( d j 1 k + m j 1 , i 1 ) ⋅ p )) = 1 + h 0 ( C , L ( − ( d j 1 k + m j 1 , i 1 + 1 ) ⋅ p )) , so that the vanishing order was exactly d j 1 k + m j 1 , i 1 d_{j_1}k+m_{j_1,i_1} d j 1 k + m j 1 , i 1
Instead of the condition corresponding to ( j 2 , i 2 ) (j_2,i_2) ( j 2 , i 2 ) Proposition 2.2
rk [ ( V j 2 / Im j 2 ( V j 2 − 1 ) ) ⟶ H 0 ( C , L ( − d j 2 k ⋅ p ) ∣ m j 2 , i 2 ⋅ p ) / φ j 2 , i 2 ( Im j 2 ( V j 2 − 1 ) ) ] ≤ i 2 − 1. \operatorname{rk}\left[(V_{j_2}/\operatorname{Im}_{j_2}(V_{j_2-1}))\longrightarrow H^0(C,L(-d_{j_2}k\cdot p)|_{m_{j_2,i_2}\cdot p})/\varphi_{j_2,i_2}(\operatorname{Im}_{j_2}(V_{j_2-1}))\right]\leq i_2-1. rk [ ( V j 2 / Im j 2 ( V j 2 − 1 )) ⟶ H 0 ( C , L ( − d j 2 k ⋅ p ) ∣ m j 2 , i 2 ⋅ p ) / φ j 2 , i 2 ( Im j 2 ( V j 2 − 1 )) ] ≤ i 2 − 1. we impose the condition
rk [ ( V j 2 / Im j 2 ( V j 2 − 1 ) ) ⟶ H 0 ( C , L ( − d j 2 k ⋅ p ) ∣ m j 1 , i 1 ⋅ p ) / φ j 2 , i 2 ( Im j 2 ( V j 2 − 1 ) ) ] ≤ i 2 − 1. \operatorname{rk}\left[(V_{j_2}/\operatorname{Im}_{j_2}(V_{j_2-1}))\longrightarrow H^0(C,L(-d_{j_2}k\cdot p)|_{m_{j_1,i_1}\cdot p})/\varphi_{j_2,i_2}(\operatorname{Im}_{j_2}(V_{j_2-1}))\right]\leq i_2-1. rk [ ( V j 2 / Im j 2 ( V j 2 − 1 )) ⟶ H 0 ( C , L ( − d j 2 k ⋅ p ) ∣ m j 1 , i 1 ⋅ p ) / φ j 2 , i 2 ( Im j 2 ( V j 2 − 1 )) ] ≤ i 2 − 1. We then restrict again to the locus of ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e )
d j 2 k + m j 1 , i 1 and d j 2 k + m j 1 , i 1 + 1 d_{j_2}k+m_{j_1,i_1}\qquad\text{and}\qquad d_{j_2}k+m_{j_1,i_1}+1 d j 2 k + m j 1 , i 1 and d j 2 k + m j 1 , i 1 + 1 are vanishing orders.
We look at the vector bundle whose fiber over each ( L , V 1 , … , V s ) (L,V_1,\ldots,V_s) ( L , V 1 , … , V s )
H 0 ( C , L ( − d j 2 k ⋅ p ) ) m j 1 , i 1 p / φ j 2 , i 2 ( Im j 2 ( V j 2 − 1 ) ) . H^0\left(C,L(-d_{j_2}k\cdot p)\right)_{m_{j_1,i_1}p}/\varphi_{j_2,i_2}(\operatorname{Im}_{j_2}(V_{j_2-1})). H 0 ( C , L ( − d j 2 k ⋅ p ) ) m j 1 , i 1 p / φ j 2 , i 2 ( Im j 2 ( V j 2 − 1 )) . This has rank i − # { m ∈ M 1 ∪ M 2 ∪ ⋯ ∪ M j 2 − 1 ∣ m < i } i-\#\{m\in M_1\cup M_2\cup\cdots\cup M_{j_2-1}\mid m<i\} i − # { m ∈ M 1 ∪ M 2 ∪ ⋯ ∪ M j 2 − 1 ∣ m < i } d j 1 k + m j 1 , i 1 d_{j_1}k+m_{j_1,i_1} d j 1 k + m j 1 , i 1 d j 1 k + m j 1 , i 1 + 1 d_{j_1}k+m_{j_1,i_1}+1 d j 1 k + m j 1 , i 1 + 1 L L L
F j 2 , i 2 / φ j 2 , i 2 ( Im j 2 ( V j 2 − 1 ) ) F_{j_2,i_2}/\varphi_{j_2,i_2}(\operatorname{Im}_{j_2}(V_{j_2-1})) F j 2 , i 2 / φ j 2 , i 2 ( Im j 2 ( V j 2 − 1 )) from the first proof of Proposition Proposition 2.2 , and we know that this has rank
i + 1 − # { m ∈ M 1 ∪ M 2 ∪ ⋯ ∪ M j 2 − 1 ∣ m < i + 1 } . i+1-\#\{m\in M_1\cup M_2\cup\cdots\cup M_{j_2-1}\mid m<i+1\}. i + 1 − # { m ∈ M 1 ∪ M 2 ∪ ⋯ ∪ M j 2 − 1 ∣ m < i + 1 } . Since i ∈ M j 1 i\in M_{j_1} i ∈ M j 1 Proposition 2.2 .
All flag codimension bounds are therefore unchanged. As in the first proof of Proposition 2.2 , successive flag degeneracy conditions followed by open restrictions show that every irreducible component of the resulting union has dimension at least g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec{e})-\operatorname{inv}(\vec{e}) g − u ( e ) − inv ( e ) W e ⃗ ( π , p ) ∪ W f ⃗ ( π , p ) W^{\vec{e}}(\pi,p)\cup W^{\vec{f}}(\pi,p) W e ( π , p ) ∪ W f ( π , p )
Except for the two vanishing and rank conditions outlined above, all the other condition stay the same. In particular, all vanishing orders in a ( e ⃗ ) a(\vec{e}) a ( e ) i i i i + 1 i+1 i + 1 k k k
The rest of the vanishing orders depend on whether d j 1 k + m j 1 , i 1 d_{j_1}k+m_{j_1,i_1} d j 1 k + m j 1 , i 1 d j 1 k + m j 1 , i 1 + 1 d_{j_1}k+m_{j_1,i_1}+1 d j 1 k + m j 1 , i 1 + 1 L L L W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) W f ⃗ ( π , p ) W^{\vec{f}}(\pi,p) W f ( π , p )
Since W f ⃗ ( π , p ) W^{\vec{f}}(\pi,p) W f ( π , p ) g − 1 − u ( e ⃗ ) − inv ( e ⃗ ) g-1-u(\vec{e})-\operatorname{inv}(\vec{e}) g − 1 − u ( e ) − inv ( e ) W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p )
To get the statement when e k − 1 + 1 > e 0 e_{k-1}+1>e_0 e k − 1 + 1 > e 0
f ⃗ = ( e k − 1 + 1 , e 1 , … , e k − 2 , e 0 − 1 ) , \vec{f}=(e_{k-1}+1,e_1,\ldots,e_{k-2},e_0-1), f = ( e k − 1 + 1 , e 1 , … , e k − 2 , e 0 − 1 ) , we first note that since
W e ⃗ ( π , p ) = { L ∣ π ∗ ( L ( − i p ) ) = O ( σ i ( e ⃗ ) ) for 0 ≤ i ≤ k − 1 (eq. for all i ) } , W^{\vec{e}}(\pi,p)=\{L\mid\pi_*(L(-ip))=\mathcal{O}(\sigma^{i}(\vec{e}))\text{ for }0\leq i\leq k-1\text{ (eq. for all }i)\}, W e ( π , p ) = { L ∣ π ∗ ( L ( − i p )) = O ( σ i ( e )) for 0 ≤ i ≤ k − 1 (eq. for all i )} , we have
W e ⃗ ( π , p ) ⊗ O ( − i p ) = W σ i ( e ⃗ ) ( π , p ) and W ‾ e ⃗ ( π , p ) ⊗ O ( − i p ) = W ‾ σ i ( e ⃗ ) ( π , p ) , W^{\vec{e}}(\pi,p)\otimes\mathcal{O}(-ip)=W^{\sigma^{i}(\vec{e})}(\pi,p)\text{ and }\overline{W}^{\vec{e}}(\pi,p)\otimes\mathcal{O}(-ip)=\overline{W}^{\sigma^{i}(\vec{e})}(\pi,p), W e ( π , p ) ⊗ O ( − i p ) = W σ i ( e ) ( π , p ) and W e ( π , p ) ⊗ O ( − i p ) = W σ i ( e ) ( π , p ) , as twisting by O ( − i p ) \mathcal{O}(-ip) O ( − i p ) Pic d → Pic d − i \operatorname{Pic}^{d}\to\operatorname{Pic}^{d-i} Pic d → Pic d − i L ∈ W e ⃗ ( π , p ) L\in W^{\vec{e}}(\pi,p) L ∈ W e ( π , p ) O ( − p ) \mathcal{O}(-p) O ( − p )
W ( e 2 , … , e k − 2 , e 0 − 1 , e k − 1 ) ⊆ W ‾ ( e 2 , … , e k − 2 , e k − 1 , e 0 − 1 ) W^{(e_2,\ldots,e_{k-2},e_0-1,e_{k-1})}\subseteq\overline{W}^{(e_2,\ldots,e_{k-2},e_{k-1},e_0-1)} W ( e 2 , … , e k − 2 , e 0 − 1 , e k − 1 ) ⊆ W ( e 2 , … , e k − 2 , e k − 1 , e 0 − 1 ) and twisting up by O ( p ) \mathcal{O}(p) O ( p ) □ \square □
Hurwitz–Brill–Noether loci
In fact, Theorem 2.5 generalizes the dimension result in [15 ] . As a consequence of Theorem 2.5 and Proposition 2.7 we obtain the following.
Corollary 2.8. Let [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^{1},p] [ π : C → P 1 , p ] H g , k t o t H^{\mathrm{tot}}_{g,k} H g , k tot e ⃗ ∈ Z ≥ 0 k \vec{e}\in\mathbb{Z}^{k}_{\geq0} e ∈ Z ≥ 0 k
W ord ( e ⃗ ) ( π , p ) and { L ∈ Pic ( C ) ∣ π ∗ L ≅ O ( e ⃗ ) } , W^{\operatorname{ord}(\vec{e})}(\pi,p)\quad\text{and}\quad\{L\in\operatorname{Pic}(C)\mid\pi_*L\cong\mathcal{O}(\vec{e})\}, W ord ( e ) ( π , p ) and { L ∈ Pic ( C ) ∣ π ∗ L ≅ O ( e )} , are birationally equivalent of dimension g − u ( e ⃗ ) g-u(\vec{e}) g − u ( e )
Proof. Using Proposition 2.7 we obtain
W e ⃗ ( π , p ) ⊆ W ord ( e ⃗ ) ( π , p ) ‾ W^{\vec{e}}(\pi,p) \subseteq\overline{W^{\operatorname{ord}(\vec{e})}(\pi,p)} W e ( π , p ) ⊆ W ord ( e ) ( π , p ) for every vector e ⃗ \vec{e} e
{ L ∈ Pic ( C ) ∣ π ∗ L ≅ O ( e ⃗ ) } = ⋃ f ⃗ a reordering of e ⃗ W f ⃗ ( π , p ) . \{L \in\operatorname{Pic}(C) \mid\pi_*L \cong\mathcal{O}(\vec{e})\}
=
\bigcup_{\vec{f}\text{ a reordering of }\vec{e}} W^{\vec{f}}(\pi,p). { L ∈ Pic ( C ) ∣ π ∗ L ≅ O ( e )} = f a reordering of e ⋃ W f ( π , p ) . In particular, the spaces
{ L ∈ Pic ( C ) ∣ π ∗ L ≅ O ( e ⃗ ) } and W ord ( e ⃗ ) ( π , p ) \{L \in\operatorname{Pic}(C) \mid\pi_*L \cong\mathcal{O}(\vec{e})\}
\quad\text{and}\quad
W^{\operatorname{ord}(\vec{e})}(\pi,p) { L ∈ Pic ( C ) ∣ π ∗ L ≅ O ( e )} and W ord ( e ) ( π , p ) have the same closure in Pic ( C ) \operatorname{Pic}(C) Pic ( C )
Corollary 2.10. Let [ π : C → P 1 , p 1 , p 2 , … , p k ] [\pi:C\to\mathbb{P}^{1},p_{1},p_{2},\ldots,p_{k}] [ π : C → P 1 , p 1 , p 2 , … , p k ] k : 1 k:1 k : 1 k k k e ⃗ ∈ Z ≥ 0 k \vec{e}\in\mathbb{Z}^{k}_{\geq0} e ∈ Z ≥ 0 k
W ord ( e ⃗ ) ( π , p 1 , p 2 , … , p k ) W^{\operatorname{ord}(\vec{e})}(\pi,p_{1},p_{2},\ldots,p_{k}) W ord ( e ) ( π , p 1 , p 2 , … , p k ) has dimension g − u ( e ⃗ ) g-u(\vec{e}) g − u ( e )
Consequently, we recover the dimension result of [15 ] .
It is a very natural question to ask, what loci are contained in the closure of W e ⃗ ( π , p ) ‾ \overline{W^{\vec{e}}(\pi,p)} W e ( π , p ) W f ⃗ ( π , p ) ‾ ⊆ W e ⃗ ( π , p ) ‾ \overline{W^{\vec{f}}(\pi,p)}\subseteq\overline{W^{\vec{e}}(\pi,p)} W f ( π , p ) ⊆ W e ( π , p ) a ( e ⃗ ) ≤ a ( f ⃗ ) \mathbf{a}(\vec{e})\leq\mathbf{a}(\vec{f}) a ( e ) ≤ a ( f )
Proposition 2.11. Let [ π : C → P 1 ] [\pi:C\to\mathbb{P}^{1}] [ π : C → P 1 ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot e ⃗ \vec{e} e k k k
⋂ i = 0 k − 1 { L ∈ Pic ( C ) ∣ π ∗ ( L − ( i ⋅ p ) ) ≅ O ( σ i e ⃗ ) } = W e ⃗ ( π , p ) ‾ ∪ Δ , \bigcap_{i=0}^{k-1}\{L\in\operatorname{Pic}(C)\mid\pi_*(L-(i\cdot p))\cong\mathcal{O}(\sigma^{i}\vec{e})\}
=
\overline{W^{\vec{e}}(\pi,p)}\cup\Delta, i = 0 ⋂ k − 1 { L ∈ Pic ( C ) ∣ π ∗ ( L − ( i ⋅ p )) ≅ O ( σ i e )} = W e ( π , p ) ∪ Δ , where Δ \Delta Δ g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec{e})-\operatorname{inv}(\vec{e}) g − u ( e ) − inv ( e )
Proof. We know from Proposition 1.10 that
W e ⃗ ( π , p ) ‾ = ⋂ i = 0 k − 1 { L ∈ Pic ( C ) ∣ π ∗ ( L − ( i ⋅ p ) ) ≅ O ( σ i e ⃗ ) } , \overline{W^{\vec{e}}(\pi,p)}
=
\bigcap_{i=0}^{k-1}\{L\in\operatorname{Pic}(C)\mid\pi_*(L-(i\cdot p))\cong\mathcal{O}(\sigma^{i}\vec{e})\}, W e ( π , p ) = i = 0 ⋂ k − 1 { L ∈ Pic ( C ) ∣ π ∗ ( L − ( i ⋅ p )) ≅ O ( σ i e )} , and hence W e ⃗ ( π , p ) ‾ \overline{W^{\vec{e}}(\pi,p)} W e ( π , p ) Δ ′ \Delta' Δ ′ a ( f ⃗ ) \mathbf{a}(\vec{f}) a ( f ) p p p L L L Δ ′ \Delta' Δ ′
We have that a ( e ⃗ ) ≤ a ( f ⃗ ) \mathbf{a}(\vec{e})\leq\mathbf{a}(\vec{f}) a ( e ) ≤ a ( f ) Δ ′ ≠ W e ⃗ ( π , p ) ‾ \Delta'\neq\overline{W^{\vec{e}}(\pi,p)} Δ ′ = W e ( π , p ) Proposition 1.18 and Theorem 2.5 , we obtain
dim ( Δ ′ ) ≤ dim ( W f ⃗ ( π , p ) ‾ ) = g − u ( f ⃗ ) − inv ( f ⃗ ) < g − u ( e ⃗ ) − inv ( e ⃗ ) . \dim(\Delta')\leq\dim(\overline{W^{\vec{f}}(\pi,p)})=g-u(\vec{f})-\operatorname{inv}(\vec{f})<g-u(\vec{e})-\operatorname{inv}(\vec{e}). dim ( Δ ′ ) ≤ dim ( W f ( π , p ) ) = g − u ( f ) − inv ( f ) < g − u ( e ) − inv ( e ) . In fact, we will see in Proposition 3.8 that Δ = ∅ \Delta= \varnothing Δ = ∅
Scheme-theoretic regeneration
The goal of this section is to prove set-theoretic, and scheme-theoretic regeneration, in the spirit of [6 Theorem 3.4] and [15 Section 6 and Section 7] . This will require several new definitions along with a description of our set-up: We start by looking at the limit linear series in Lemma 1.12 and describe what ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e )
Nested limit linear series
To describe ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) Lemma 1.12 .
Lemma 3.1. Let π : X → P 1 \pi:X\to\mathbb{P}^{1} π : X → P 1 k : 1 k:1 k : 1 q ∈ X q\in X q ∈ X e ⃗ = ( e 0 , … , e k − 1 ) \vec{e}=(e_{0},\ldots,e_{k-1}) e = ( e 0 , … , e k − 1 ) k k k e 0 > e k − 1 + 1 e_{0}>e_{k-1}+1 e 0 > e k − 1 + 1 L L L
W t 0 ( e ⃗ ) ( π , q ) ⊗ O X ( q ) . W^{t_{0}(\vec{e})}(\pi,q)\otimes\mathcal{O}_{X}(q). W t 0 ( e ) ( π , q ) ⊗ O X ( q ) . Then L L L ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) V V V
dim V ( − 2 q ) = dim V − 1. \dim V(-2q)=\dim V-1. dim V ( − 2 q ) = dim V − 1. Proof. We consider the vector
f ⃗ = ( e 0 , e k − 1 + 1 , e 1 , … , e k − 2 ) , \vec{f}=(e_{0},e_{k-1}+1,e_{1},\ldots,e_{k-2}), f = ( e 0 , e k − 1 + 1 , e 1 , … , e k − 2 ) , and we immediately see
W t 0 ( e ⃗ ) ( π , q ) ⊗ O X ( q ) = W f ⃗ ( π , q ) . W^{t_{0}(\vec{e})}(\pi,q)\otimes\mathcal{O}_{X}(q)=W^{\vec{f}}(\pi,q). W t 0 ( e ) ( π , q ) ⊗ O X ( q ) = W f ( π , q ) . To determine the vanishing orders of V V V q q q a ( e ⃗ ) \mathfrak{a}(\vec{e}) a ( e ) V V V
h 0 ( X , L ) = ∑ i = 0 k − 1 ( f i + 1 ) = ∑ i = 0 k − 1 ( e i + 1 ) + 1 , h^{0}(X,L)=\sum_{i=0}^{k-1}(f_{i}+1)=\sum_{i=0}^{k-1}(e_{i}+1)+1, h 0 ( X , L ) = i = 0 ∑ k − 1 ( f i + 1 ) = i = 0 ∑ k − 1 ( e i + 1 ) + 1 , hence dim V = h 0 ( C , L ) − 1 \dim V=h^{0}(C,L)-1 dim V = h 0 ( C , L ) − 1
dim V ( − 2 q ) = dim V − 1 , \dim V(-2q)=\dim V-1, dim V ( − 2 q ) = dim V − 1 , it follows that either 0 or 1 is not a vanishing order, and V ( − 2 q ) = H 0 ( C , L − 2 q ) V(-2q)=H^{0}(C,L-2q) V ( − 2 q ) = H 0 ( C , L − 2 q ) a ( f ⃗ ) ≥ 2 \mathfrak{a}(\vec{f})_{\geq2} a ( f ) ≥ 2 V V V
In particular, the integer e 0 k e_{0}k e 0 k V V V V V V q q q
V ( − e 0 k ⋅ q ) ⊗ H 0 ( P 1 , O P 1 ( e 0 ) ) ⟶ V , V(-e_{0}k\cdot q)\otimes H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(e_{0}))\longrightarrow V, V ( − e 0 k ⋅ q ) ⊗ H 0 ( P 1 , O P 1 ( e 0 )) ⟶ V , and we will construct a section in the image with vanishing order 0 at q q q σ ∈ V \sigma\in V σ ∈ V ord q ( σ ) = e 0 k \operatorname{ord}_{q}(\sigma)=e_{0}k ord q ( σ ) = e 0 k σ \sigma σ L ( − e 0 k ⋅ q ) L(-e_{0}k\cdot q) L ( − e 0 k ⋅ q ) s ∈ H 0 ( P 1 , O P 1 ( 1 ) ) s\in H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(1)) s ∈ H 0 ( P 1 , O P 1 ( 1 )) q q q P 1 \mathbb{P}^{1} P 1 σ ⋅ s e 0 \sigma\cdot s^{e_{0}} σ ⋅ s e 0 ord q ( σ ⋅ s e 0 ) = 0 \operatorname{ord}_{q}(\sigma\cdot s^{e_{0}})=0 ord q ( σ ⋅ s e 0 ) = 0
It follows that 0 is a vanishing order of V V V
V = ⟨ H 0 ( C , L ( − 2 q ) ) , σ ⋅ s e 0 ⟩ . V=\langle H^{0}(C,L(-2q)),\sigma\cdot s^{e_{0}}\rangle. V = ⟨ H 0 ( C , L ( − 2 q )) , σ ⋅ s e 0 ⟩ . It remains to show that this space is uniquely determined. Indeed, suppose that σ ′ \sigma' σ ′
ord q ( σ ′ ) = e 0 k . \operatorname{ord}_{q}(\sigma')=e_{0}k. ord q ( σ ′ ) = e 0 k . Since k e 0 + 1 ke_{0}+1 k e 0 + 1 V V V t ∈ C t\in\mathbb{C} t ∈ C
ord q ( σ − t σ ′ ) ≥ e 0 k + 2. \operatorname{ord}_{q}(\sigma-t\sigma')\geq e_{0}k+2. ord q ( σ − t σ ′ ) ≥ e 0 k + 2. Therefore,
( σ − t σ ′ ) ⋅ s e 0 ∈ H 0 ( X , L ( − 2 q ) ) , (\sigma-t\sigma')\cdot s^{e_0}\in H^0(X,L(-2q)), ( σ − t σ ′ ) ⋅ s e 0 ∈ H 0 ( X , L ( − 2 q )) , so replacing σ \sigma σ σ ′ \sigma' σ ′ V V V
V i = V ( − d i k ⋅ q ) : = H 0 ( X , L − d i k ⋅ q ) ∩ V V_i=V(-d_{i k}\cdot q):=H^0(X,L-d_{i k}\cdot q)\cap V V i = V ( − d ik ⋅ q ) := H 0 ( X , L − d ik ⋅ q ) ∩ V for all 1 ≤ i ≤ s 1\leq i\leq s 1 ≤ i ≤ s d i > e k − 1 + 1 d_i>e_{k-1}+1 d i > e k − 1 + 1
V i = ⟨ V ( − ( d i k + 2 ) ⋅ q ) , σ ⋅ s e 0 − d i ⟩ V_i=\left\langle V(-(d_i k+2)\cdot q),\ \sigma\cdot s^{e_0-d_i}\right\rangle V i = ⟨ V ( − ( d i k + 2 ) ⋅ q ) , σ ⋅ s e 0 − d i ⟩ for d i ≤ e k − 1 + 1 d_i\leq e_{k-1}+1 d i ≤ e k − 1 + 1
This result extends for all other loci appearing in Lemma 1.12 . In these cases, describing the nested linear series is straight-forward, as the condition dim V ( − q ) = dim V \dim V(-q)=\dim V dim V ( − q ) = dim V V = H 0 ( X , L − q ) V=H^0(X,L-q) V = H 0 ( X , L − q )
Lemma 3.2. Let π : X → P 1 \pi:X\to\mathbb{P}^1 π : X → P 1 k k k q q q e ⃗ ∈ Z ≥ 0 k \vec{e}\in\mathbb{Z}_{\geq0}^k e ∈ Z ≥ 0 k
L ∈ W e ⃗ ( π , q ) ⊗ O X ( q ) or L ∈ W t i ( e ⃗ ) ( π , q ) ⊗ O X ( q ) , 1 ≤ i ≤ k − 1. L\in W^{\vec{e}}(\pi,q)\otimes\mathcal{O}_X(q)\quad\text{or}\quad L\in W^{t_i(\vec{e})}(\pi,q)\otimes\mathcal{O}_X(q),\qquad1\leq i\leq k-1. L ∈ W e ( π , q ) ⊗ O X ( q ) or L ∈ W t i ( e ) ( π , q ) ⊗ O X ( q ) , 1 ≤ i ≤ k − 1. Then L L L ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) V ( − q ) = V V(-q)=V V ( − q ) = V
V = H 0 ( X , L ( − q ) ) , V j = H 0 ( X , L ( − ( d j k + 1 ) q ) ) . V=H^0(X,L(-q)),\qquad V_j=H^0(X,L(-(d_jk+1)q)). V = H 0 ( X , L ( − q )) , V j = H 0 ( X , L ( − ( d j k + 1 ) q )) . The results above will be used to describe the ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) Lemma 1.12 . We will prove a regeneration formula for the nested linear series in Lemma 3.1 and Lemma 3.2 .
Definition 3.3. Let [ π 0 , p ] [\pi_0,p] [ π 0 , p ] G ord ( e ⃗ ) , n c ( π 0 , p ) G^{\operatorname{ord}(\vec{e}),\mathrm{nc}}(\pi_0,p) G ord ( e ) , nc ( π 0 , p ) ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) q q q X X X p , q p,q p , q E E E
l E = ( L E , V E ) , l X = ( L X , V X ) l_E=(L_E,V_E),\qquad l_X=(L_X,V_X) l E = ( L E , V E ) , l X = ( L X , V X ) satisfy a ( V E , p ) = a ( e ⃗ ) a(V_E,p)=a(\vec{e}) a ( V E , p ) = a ( e )
L X ( − q ) ∈ W e ⃗ ( π X , q ) L_X(-q)\in W^{\vec{e}}(\pi_X,q) L X ( − q ) ∈ W e ( π X , q ) a ( V X , q ) = 1 + a ( e ⃗ ) a(V_X,q)=1+a(\vec{e}) a ( V X , q ) = 1 + a ( e )
For some 1 ≤ j ≤ k − 1 1\leq j\leq k-1 1 ≤ j ≤ k − 1 t j ( e ⃗ ) ≠ e ⃗ t_j(\vec{e})\ne\vec{e} t j ( e ) = e
L E ≃ O E ( j p + ( d − j ) q ) , L X ( − q ) ∈ W t j ( e ⃗ ) ( π X , q ) , L_E\simeq\mathcal{O}_E(jp+(d-j)q),\qquad L_X(-q)\in W^{t_j(\vec{e})}(\pi_X,q), L E ≃ O E ( j p + ( d − j ) q ) , L X ( − q ) ∈ W t j ( e ) ( π X , q ) , and a ( V X , q ) = 1 + a ( t j ( e ⃗ ) ) a(V_X,q)=1+a(t_j(\vec{e})) a ( V X , q ) = 1 + a ( t j ( e ))
If t 0 ( e ⃗ ) ≠ e ⃗ t_0(\vec{e})\ne\vec{e} t 0 ( e ) = e
L E ≃ O E ( d q ) , L X ( − q ) ∈ W t 0 ( e ⃗ ) ( π X , q ) , L_E\simeq\mathcal{O}_E(dq),\qquad L_X(-q)\in W^{t_0(\vec{e})}(\pi_X,q), L E ≃ O E ( d q ) , L X ( − q ) ∈ W t 0 ( e ) ( π X , q ) , and
a ( V X , q ) = a ( f ⃗ ) ∖ { 1 } , where f ⃗ = ( e 0 , e k − 1 + 1 , e 1 , … , e k − 2 ) . a(V_X,q)=a(\vec{f})\setminus\{1\},\qquad\text{where }\vec{f}=(e_0,e_{k-1}+1,e_1,\ldots,e_{k-2}). a ( V X , q ) = a ( f ) ∖ { 1 } , where f = ( e 0 , e k − 1 + 1 , e 1 , … , e k − 2 ) . Here 1 + a 1+a 1 + a
The layers of every series in this locus are uniquely determined by its untwisted aspects. On E E E
V E , i = V E ( − d i k p ) , V_{E,i}=V_E(-d_{ik}p), V E , i = V E ( − d ik p ) , since nesting gives containment and the prescribed vanishing sequence at p p p Lemma 3.2 gives
V X = H 0 ( X , L X ( − q ) ) , V X , i = H 0 ( X , L X ( − ( d i k + 1 ) q ) ) . V_X=H^0(X,L_X(-q)),\qquad V_{X,i}=H^0(X,L_X(-(d_{ik}+1)q)). V X = H 0 ( X , L X ( − q )) , V X , i = H 0 ( X , L X ( − ( d ik + 1 ) q )) . In case (3), they are the unique spaces constructed in Lemma 3.1 .
Set-theoretic regeneration
Setup 3.1. Fix the admissible cover
π 0 : X ∪ q E ⟶ P X ∪ q P E , p ∈ E \pi_0 : X \mathbin{\cup_q} E \longrightarrow P_X \mathbin{\cup_q} P_E,\qquad p \in E π 0 : X ∪ q E ⟶ P X ∪ q P E , p ∈ E described in Setup 1.1. By the deformation theory of Hurwitz spaces and admissible covers, see [11 , 1 ] , there exists a smoothing family
C → π P p ↓ ↓ B = B \begin{CD}
C @>{\pi}>> P \\
@V{p}VV @VV{}V \\
B @= B
\end{CD} C p ↓ ⏐ B π P ↓ ⏐ B over B = Spec C [ [ t ] ] B = \operatorname{Spec}\mathbb{C}[[t]] B = Spec C [[ t ]] ( π 0 , p ) (\pi_0,p) ( π 0 , p ) k k k p p p C C C P → B P \to B P → B
Let e ⃗ \vec{e} e k k k [15 Theorem 6.5] to obtain a quasi-projective scheme G e ⃗ , n c ( π , p ) \mathcal{G}^{\vec{e},\mathrm{nc}}(\pi,p) G e , nc ( π , p ) t ≠ 0 t \ne0 t = 0 e ⃗ \vec{e} e e ⃗ \vec{e} e
Theorem 3.4 (Regeneration). Let e ⃗ \vec{e} e k k k C → P → B C \to P \to B C → P → B B B B G e ⃗ , n c ( π , p ) → B \mathcal{G}^{\vec{e},\mathrm{nc}}(\pi,p) \to B G e , nc ( π , p ) → B e ⃗ \vec{e} e a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) p p p G e ⃗ , n c ( π 0 , p ) \mathcal{G}^{\vec{e},\mathrm{nc}}(\pi_0,p) G e , nc ( π 0 , p ) Z Z Z G e ⃗ , n c ( π , p ) \mathcal{G}^{\vec{e},\mathrm{nc}}(\pi,p) G e , nc ( π , p )
dim Z ≥ dim Pic ( C / B ) − u ( e ⃗ ) − inv ( e ⃗ ) . \dim Z \ge\dim\operatorname{Pic}(C/B) - u(\vec{e}) - \operatorname{inv}(\vec{e}). dim Z ≥ dim Pic ( C / B ) − u ( e ) − inv ( e ) . Proof. The proof of [15 Theorem 6.5] applies almost verbatim to construct a quasi-projective scheme G ord ( e ⃗ ) , n c ( π , p ) \mathcal{G}^{\operatorname{ord}(\vec{e}),\mathrm{nc}}(\pi,p) G ord ( e ) , nc ( π , p ) ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) Pic ( C / B ) − u ( e ⃗ ) \operatorname{Pic}(C/B) - u(\vec{e}) Pic ( C / B ) − u ( e )
The quasi-projective scheme G ord ( e ⃗ ) , n c ( π , p ) \mathcal{G}^{\operatorname{ord}(\vec{e}),\mathrm{nc}}(\pi,p) G ord ( e ) , nc ( π , p ) ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e )
G e ⃗ , n c ( π , p ) ⊆ G ord ( e ⃗ ) , n c ( π , p ) \mathcal{G}^{\vec{e},\mathrm{nc}}(\pi,p) \subseteq\mathcal{G}^{\operatorname{ord}(\vec{e}),\mathrm{nc}}(\pi,p) G e , nc ( π , p ) ⊆ G ord ( e ) , nc ( π , p ) for which the ramification orders at p p p a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) p p p G e ⃗ , n c ( π 0 , p ) \mathcal{G}^{\vec{e},\mathrm{nc}}(\pi_0,p) G e , nc ( π 0 , p )
Moreover, the proof of Proposition 2.2 implies that
codim ( G e ⃗ , n c ( π , p ) , G ord ( e ⃗ ) , n c ( π , p ) ) ≤ inv ( e ⃗ ) . \operatorname{codim}\left(\mathcal{G}^{\vec{e},\mathrm{nc}}(\pi,p),\mathcal{G}^{\operatorname{ord}(\vec{e}),\mathrm{nc}}(\pi,p)\right) \le\operatorname{inv}(\vec{e}). codim ( G e , nc ( π , p ) , G ord ( e ) , nc ( π , p ) ) ≤ inv ( e ) . This implies that every irreducible component Z Z Z G e ⃗ , n c ( π , p ) \mathcal{G}^{\vec{e},\mathrm{nc}}(\pi,p) G e , nc ( π , p )
dim Z ≥ dim Pic ( C / B ) − u ( e ⃗ ) − inv ( e ⃗ ) , \dim Z \ge\dim\operatorname{Pic}(C/B) - u(\vec{e}) - \operatorname{inv}(\vec{e}), dim Z ≥ dim Pic ( C / B ) − u ( e ) − inv ( e ) , which is what we wanted to prove.
Let d = g + ∣ e ⃗ ∣ + k − 1 d = g + |\vec{e}| + k - 1 d = g + ∣ e ∣ + k − 1 G e ⃗ , n c ( π , p ) \mathcal{G}^{\vec{e},\mathrm{nc}}(\pi,p) G e , nc ( π , p ) Pic d ( C / B ) \operatorname{Pic}^{d}(C/B) Pic d ( C / B ) Theorem 3.4 .
Theorem 3.5. Let e ⃗ \vec e e k k k C → P → B \mathcal{C} \to\mathcal{P} \to B C → P → B B B B W e ⃗ ( π , p ) → B \mathcal{W}^{\vec e}(\pi,p) \to B W e ( π , p ) → B a ( e ⃗ ) \mathbf{a}(\vec e) a ( e ) p p p Pic d ( X ) × Pic 0 ( E ) \operatorname{Pic}^{d}(X) \times\operatorname{Pic}^{0}(E) Pic d ( X ) × Pic 0 ( E )
( O X ( q ) ⊗ W ‾ e ⃗ ( π ∣ X , q ) ) × Pic 0 ( E ) \left(\mathcal{O}_{X}(q) \otimes\overline{\mathcal{W}}^{\vec e}(\pi|_{X},q)\right) \times\operatorname{Pic}^{0}(E) ( O X ( q ) ⊗ W e ( π ∣ X , q ) ) × Pic 0 ( E )
( O X ( q ) ⊗ W ‾ t i ( e ⃗ ) ( π ∣ X , q ) ) × { O E ( i ⋅ p − i ⋅ q ) } \left(\mathcal{O}_{X}(q) \otimes\overline{\mathcal{W}}^{t_i(\vec e)}(\pi|_{X},q)\right) \times\left\{\mathcal{O}_{E}(i \cdot p-i \cdot q)\right\} ( O X ( q ) ⊗ W t i ( e ) ( π ∣ X , q ) ) × { O E ( i ⋅ p − i ⋅ q ) } 0 ≤ i ≤ k − 1 0 \leq i \leq k-1 0 ≤ i ≤ k − 1 t i ( e ⃗ ) ≠ e ⃗ t_i(\vec e) \ne\vec e t i ( e ) = e Z Z Z W e ⃗ ( π , p ) \mathcal{W}^{\vec e}(\pi,p) W e ( π , p )
dim Z ≥ dim Pic ( C / B ) − u ( e ⃗ ) − inv ( e ⃗ ) . \dim Z \geq\dim\operatorname{Pic}(\mathcal{C}/B)-u(\vec e)-\operatorname{inv}(\vec e). dim Z ≥ dim Pic ( C / B ) − u ( e ) − inv ( e ) . Proof. Consider the forgetful map
G e ⃗ , n c ( π , p ) ⟶ Pic d ( C / B ) , \mathcal{G}^{\vec e,\mathrm{nc}}(\pi,p) \longrightarrow\operatorname{Pic}^{d}(\mathcal{C}/B), G e , nc ( π , p ) ⟶ Pic d ( C / B ) , whose image is an open subspace of W e ⃗ ( π , p ) \mathcal{W}^{\vec e}(\pi,p) W e ( π , p )
h 0 ( C , L ) = ∑ i = 0 k − 1 ( e i + 1 ) h^{0}(\mathcal{C},L)=\sum_{i=0}^{k-1}(e_i+1) h 0 ( C , L ) = i = 0 ∑ k − 1 ( e i + 1 ) and having vanishing orders exactly a ( e ⃗ ) \mathbf{a}(\vec e) a ( e ) ord ( e ⃗ ) \operatorname{ord}(\vec e) ord ( e )
V j = H 0 ( C , L − d j k ⋅ p ) . V_j=H^{0}(\mathcal{C},L-d_{jk}\cdot p). V j = H 0 ( C , L − d j k ⋅ p ) . Above the boundary, the forgetful map is injective because of Lemma 3.1 and Lemma 3.2 , and the conclusion follows. □
Scheme theoretic regeneration, Reduceness, and Cohen–Macaulayness
We know from Proposition 2.11 that we have the set-theoretic intersection
\bigcap_{i=0}^{k-1}\left\{L \in\operatorname{Pic}(\mathcal{C}) \mathrel{\left|} \pi_{*}(L-(i\cdot p)) \cong\mathcal{O}(\sigma^{i}\vec e)\right.\right\}=\overline{\mathcal{W}}^{\vec e}(\pi,p)\cup\Delta,
where Δ \Delta Δ dim Δ < g − u ( e ⃗ ) − inv ( e ⃗ ) \dim\Delta<g-u(\vec e)-\operatorname{inv}(\vec e) dim Δ < g − u ( e ) − inv ( e )
\mathcal{O}_{\mathcal{C}}(i\cdot p)\otimes\overline{\mathcal{W}}^{\sigma^{i}\vec e}(\pi,p)=\left\{L\in\operatorname{Pic}^{d}(\mathcal{C})\mathrel{\left|}h^{0}(\mathcal{C},L-(mk+i)p)\geq h^{0}(\mathbb{P}^{1},\mathcal{O}(\sigma^{i}\vec e)(-m))\text{ for }m\geq0\right.\right\}.
As such, using all rank conditions
\left\{L\in\operatorname{Pic}^{d}(\mathcal{C})\mathrel{\left|}h^{0}(\mathcal{C},L-(mk+i)p)\geq h^{0}(\mathbb{P}^{1},\mathcal{O}(\sigma^{i}\vec e)(-m))\text{ for }m\geq0,\text{ and }0\leq i\leq k-1\right.\right\},
the space W ‾ e ⃗ ( π , p ) ∪ Δ \overline{\mathcal{W}}^{\vec e}(\pi,p)\cup\Delta W e ( π , p ) ∪ Δ g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec e)-\operatorname{inv}(\vec e) g − u ( e ) − inv ( e ) Proposition 2.11 , this implies that Δ = ∅ \Delta=\varnothing Δ = ∅ W ‾ e ⃗ ( π , p ) \overline{\mathcal{W}}^{\vec e}(\pi,p) W e ( π , p )
The same methods as in [15 Section 7] can be applied for the elliptic tail degeneration. Let C 0 = X ∪ q E \mathcal{C}_{0}=X\cup_{q}E C 0 = X ∪ q E X X X E E E g − 1 g-1 g − 1 1 1 1
d ⃗ = ( d X , d E ) , d X + d E = d , \vec d=(d_X,d_E),\qquad d_X+d_E=d, d = ( d X , d E ) , d X + d E = d , and consider the determinantal locus
\mathcal{W}^{r}_{\vec d}(\mathcal{C}_{0})=\left\{L\in\operatorname{Pic}^{\vec d}(\mathcal{C}_{0})\mathrel{\left|}h^{0}(\mathcal{C}_{0},L)\geq r+1\right.\right\}.
Since C 0 C_0 C 0
Pic d ⃗ ( C 0 ) ≃ Pic d X ( X ) × Pic d E ( E ) , \operatorname{Pic}^{\vec d}(C_0) \simeq\operatorname{Pic}^{d^X}(X) \times\operatorname{Pic}^{d^E}(E), Pic d ( C 0 ) ≃ Pic d X ( X ) × Pic d E ( E ) , and our goal is to describe this rank degeneracy locus on this product. Let
π : C 0 × Pic d ⃗ ( C 0 ) ⟶ Pic d ⃗ ( C 0 ) \pi:C_0 \times\operatorname{Pic}^{\vec d}(C_0) \longrightarrow\operatorname{Pic}^{\vec d}(C_0) π : C 0 × Pic d ( C 0 ) ⟶ Pic d ( C 0 ) be the projection, and let L : = L d ⃗ \mathcal L := \mathcal L_{\vec d} L := L d N N N D D D C 0 C_0 C 0
D X : = D ∩ X and D E : = D ∩ E D_X := D \cap X \qquad\text{and} \qquad D_E := D \cap E D X := D ∩ X and D E := D ∩ E are sufficiently positive, with degrees denoted N X N_X N X N E N_E N E
ϕ : π ∗ L ( D ) ⟶ π ∗ ( L ( D ) ∣ D ) , \phi:\pi_*\mathcal L(D) \longrightarrow\pi_*(\mathcal L(D)|_D), ϕ : π ∗ L ( D ) ⟶ π ∗ ( L ( D ) ∣ D ) , the locus of interest is
W r d ⃗ ( C 0 ) = { L ∈ Pic d ⃗ ( C 0 ) | dim ker ( ϕ ∣ L ) ≥ r + 1 } . W^{\vec d}_r(C_0)=\left\{L\in\operatorname{Pic}^{\vec d}(C_0)\,\middle|\,\dim\ker(\phi|_L)\ge r+1\right\}. W r d ( C 0 ) = { L ∈ Pic d ( C 0 ) dim ker ( ϕ ∣ L ) ≥ r + 1 } . Its scheme structure is given by the ( n + 1 ) × ( n + 1 ) (n+1)\times(n+1) ( n + 1 ) × ( n + 1 ) ϕ \phi ϕ
n = rank π ∗ L ( D ) − ( r + 1 ) = d − g − r + deg ( D ) . n=\operatorname{rank}\pi_*\mathcal L(D)-(r+1)=d-g-r+\deg(D). n = rank π ∗ L ( D ) − ( r + 1 ) = d − g − r + deg ( D ) . We describe the rank loci of ϕ \phi ϕ
ν : X ⊔ E ⟶ C 0 . \nu:X\sqcup E\longrightarrow C_0. ν : X ⊔ E ⟶ C 0 . Let
π X : X × Pic d X ( X ) ⟶ Pic d X ( X ) , π E : E × Pic d E ( E ) ⟶ Pic d E ( E ) \pi_X:X\times\operatorname{Pic}^{d^X}(X)\longrightarrow\operatorname{Pic}^{d^X}(X),\qquad
\pi_E:E\times\operatorname{Pic}^{d^E}(E)\longrightarrow\operatorname{Pic}^{d^E}(E) π X : X × Pic d X ( X ) ⟶ Pic d X ( X ) , π E : E × Pic d E ( E ) ⟶ Pic d E ( E ) be the projections, and let L X \mathcal L_X L X L E \mathcal L_E L E L \mathcal L L
pr X : Pic d ⃗ ( C 0 ) ⟶ Pic d X ( X ) and pr E : Pic d ⃗ ( C 0 ) ⟶ Pic d E ( E ) \operatorname{pr}_X:\operatorname{Pic}^{\vec d}(C_0)\longrightarrow\operatorname{Pic}^{d^X}(X)
\qquad\text{and}\qquad
\operatorname{pr}_E:\operatorname{Pic}^{\vec d}(C_0)\longrightarrow\operatorname{Pic}^{d^E}(E) pr X : Pic d ( C 0 ) ⟶ Pic d X ( X ) and pr E : Pic d ( C 0 ) ⟶ Pic d E ( E ) we have
L ∣ X × Pic d ⃗ ( C 0 ) = ( Id X × pr X ) ∗ L X and L ∣ E × Pic d ⃗ ( C 0 ) = ( Id E × pr E ) ∗ L E . \mathcal L|_{X\times\operatorname{Pic}^{\vec d}(C_0)}
=(\operatorname{Id}_X\times\operatorname{pr}_X)^*\mathcal L_X
\qquad\text{and}\qquad
\mathcal L|_{E\times\operatorname{Pic}^{\vec d}(C_0)}
=(\operatorname{Id}_E\times\operatorname{pr}_E)^*\mathcal L_E. L ∣ X × Pic d ( C 0 ) = ( Id X × pr X ) ∗ L X and L ∣ E × Pic d ( C 0 ) = ( Id E × pr E ) ∗ L E . The normalization sequence gives a commutative diagram
( pr X ) ∗ ( π X ) ∗ L X ( D X ) ⊕ ( pr E ) ∗ ( π E ) ∗ L E ( D E ) ∥ 0 ⟶ π ∗ L ( D ) ⟶ π ∗ ( ν × Id ) ∗ ( ν × Id ) ∗ L ( D ) → η π ∗ ( L ( D ) ∣ q ) ⟶ 0 ↓ ϕ ↓ ψ ↓ π ∗ ( L ( D ) ∣ D ) → ∼ π ∗ ( ν × Id ) ∗ ( ν × Id ) ∗ L ( D ) ∣ ν − 1 ( D ) ∥ ( pr X ) ∗ ( π X ) ∗ ( L X ( D X ) ∣ D X ) ⊕ ( pr E ) ∗ ( π E ) ∗ ( L E ( D E ) ∣ D E ) . \begin{aligned}
&(\operatorname{pr}_X)^*(\pi_X)_*\mathcal L_X(D_X)\oplus(\operatorname{pr}_E)^*(\pi_E)_*\mathcal L_E(D_E)\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Vert\\
&0\longrightarrow\pi_*\mathcal L(D)\longrightarrow
\pi_*(\nu\times\operatorname{Id})_*(\nu\times\operatorname{Id})^*\mathcal L(D)
\xrightarrow{\eta}\pi_*(\mathcal L(D)|_q)\longrightarrow0\\
&\qquad\qquad\qquad\quad\downarrow\phi
\qquad\qquad\qquad\qquad\downarrow\psi
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\downarrow\\
&\pi_*(\mathcal L(D)|_D)\xrightarrow{\ \sim\ }
\pi_*(\nu\times\operatorname{Id})_*(\nu\times\operatorname{Id})^*\mathcal L(D)|_{\nu^{-1}(D)}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Vert\\
&(\operatorname{pr}_X)^*(\pi_X)_*(\mathcal L_X(D_X)|_{D_X})
\oplus(\operatorname{pr}_E)^*(\pi_E)_*(\mathcal L_E(D_E)|_{D_E}).
\end{aligned} ( pr X ) ∗ ( π X ) ∗ L X ( D X ) ⊕ ( pr E ) ∗ ( π E ) ∗ L E ( D E ) ∥ 0 ⟶ π ∗ L ( D ) ⟶ π ∗ ( ν × Id ) ∗ ( ν × Id ) ∗ L ( D ) η π ∗ ( L ( D ) ∣ q ) ⟶ 0 ↓ ϕ ↓ ψ ↓ π ∗ ( L ( D ) ∣ D ) ∼ π ∗ ( ν × Id ) ∗ ( ν × Id ) ∗ L ( D ) ∣ ν − 1 ( D ) ∥ ( pr X ) ∗ ( π X ) ∗ ( L X ( D X ) ∣ D X ) ⊕ ( pr E ) ∗ ( π E ) ∗ ( L E ( D E ) ∣ D E ) . Here η ( s X , s E ) = s X ( q ) − s E ( q ) \eta(s_X,s_E)=s_X(q)-s_E(q) η ( s X , s E ) = s X ( q ) − s E ( q ) ψ \psi ψ D X ⊔ D E D_X\sqcup D_E D X ⊔ D E D X D_X D X D E D_E D E h 1 ( C 0 , L ( D ) ) = 0 h^1(C_0,L(D))=0 h 1 ( C 0 , L ( D )) = 0 L ∈ Pic d ⃗ ( C 0 ) L\in\operatorname{Pic}^{\vec d}(C_0) L ∈ Pic d ( C 0 ) [15 Section 7.1] , since exact sequences of vector bundles split locally, the rank loci of ϕ \phi ϕ ψ ⊕ η \psi\oplus\eta ψ ⊕ η
{ L ∈ Pic d ⃗ ( C 0 ) | rank ϕ ∣ L ≤ n } = { L ∈ Pic d ⃗ ( C 0 ) | rank ( ψ ⊕ η ) ∣ L ≤ n + 1 } \left\{L\in\operatorname{Pic}^{\vec d}(C_0)\,\middle|\,\operatorname{rank}\phi|_L\le n\right\}
=
\left\{L\in\operatorname{Pic}^{\vec d}(C_0)\,\middle|\,\operatorname{rank}(\psi\oplus\eta)|_L\le n+1\right\} { L ∈ Pic d ( C 0 ) rank ϕ ∣ L ≤ n } = { L ∈ Pic d ( C 0 ) rank ( ψ ⊕ η ) ∣ L ≤ n + 1 } scheme-theoretically. Then we can apply [15 Lemma 7.1] and obtain the following result.
Lemma 3.6. Fix a multidegree d ⃗ = ( d X , d E ) \vec{d}=(d_X,d_E) d = ( d X , d E )
Pic d ⃗ ( C 0 ) ≅ Pic d X ( X ) × Pic d E ( E ) , \operatorname{Pic}^{\vec{d}}(C_0)\cong\operatorname{Pic}^{d_X}(X)\times\operatorname{Pic}^{d_E}(E), Pic d ( C 0 ) ≅ Pic d X ( X ) × Pic d E ( E ) , the determinantal locus
W d ⃗ r ( C 0 ) = { L : h 0 ( C 0 , L ) ≥ r + 1 } W^{r}_{\vec{d}}(C_0)=\{L:h^{0}(C_0,L)\ge r+1\} W d r ( C 0 ) = { L : h 0 ( C 0 , L ) ≥ r + 1 } satisfies
W d ⃗ r ( C 0 ) = ⋃ x , y ≥ 0 x + y = r + 1 ( O X ( q ) ⊗ W d X − 1 x − 1 ( X ) ) × ( O E ( q ) ⊗ W d E − 1 y − 1 ( E ) ) ∪ ⋃ x , y ≥ 0 x + y = r + 2 W d X x − 1 ( X ) × W d E y − 1 ( E ) W^{r}_{\vec{d}}(C_0)=
\bigcup_{\substack{x,y\ge0\\x+y=r+1}}
\left(\mathcal{O}_X(q)\otimes W^{x-1}_{d_X-1}(X)\right)\times
\left(\mathcal{O}_E(q)\otimes W^{y-1}_{d_E-1}(E)\right)
\cup
\bigcup_{\substack{x,y\ge0\\x+y=r+2}}
W^{x-1}_{d_X}(X)\times W^{y-1}_{d_E}(E) W d r ( C 0 ) = x , y ≥ 0 x + y = r + 1 ⋃ ( O X ( q ) ⊗ W d X − 1 x − 1 ( X ) ) × ( O E ( q ) ⊗ W d E − 1 y − 1 ( E ) ) ∪ x , y ≥ 0 x + y = r + 2 ⋃ W d X x − 1 ( X ) × W d E y − 1 ( E ) scheme-theoretically, where W a − 1 ( Z ) = Pic a ( Z ) W^{-1}_{a}(Z)=\operatorname{Pic}^{a}(Z) W a − 1 ( Z ) = Pic a ( Z )
The preceding lemma expresses a rank condition on the nodal curve as a union of products. The following identities allow us to intersect such loci scheme-theoretically.
Lemma 3.7. Let S S S T T T
X 1 ⊆ Z 1 ⊆ S and X 2 ⊆ Z 2 ⊆ S X_1\subseteq Z_1\subseteq S\quad\text{and}\quad X_2\subseteq Z_2\subseteq S X 1 ⊆ Z 1 ⊆ S and X 2 ⊆ Z 2 ⊆ S be closed subschemes.
(1) If Y ⊆ T Y\subseteq T Y ⊆ T
[ ( X 1 × T ) ∪ ( Z 1 × Y ) ] ∩ [ ( X 2 × T ) ∪ ( Z 2 × Y ) ] = [ ( X 1 ∩ X 2 ) × T ] ∪ [ ( Z 1 ∩ Z 2 ) × Y ] . \left[(X_1\times T)\cup(Z_1\times Y)\right]\cap\left[(X_2\times T)\cup(Z_2\times Y)\right]
=
\left[(X_1\cap X_2)\times T\right]\cup\left[(Z_1\cap Z_2)\times Y\right]. [ ( X 1 × T ) ∪ ( Z 1 × Y ) ] ∩ [ ( X 2 × T ) ∪ ( Z 2 × Y ) ] = [ ( X 1 ∩ X 2 ) × T ] ∪ [ ( Z 1 ∩ Z 2 ) × Y ] . (2) If Y 1 , Y 2 ⊆ T Y_1,Y_2\subseteq T Y 1 , Y 2 ⊆ T
[ ( X 1 × T ) ∪ ( Z 1 × Y 1 ) ] ∩ [ ( X 2 × T ) ∪ ( Z 2 × Y 2 ) ] = [ ( X 1 ∩ X 2 ) × T ] ∪ [ ( Z 1 ∩ X 2 ) × Y 1 ] ∪ [ ( X 1 ∩ Z 2 ) × Y 2 ] . \left[(X_1\times T)\cup(Z_1\times Y_1)\right]\cap\left[(X_2\times T)\cup(Z_2\times Y_2)\right]
=
\left[(X_1\cap X_2)\times T\right]\cup\left[(Z_1\cap X_2)\times Y_1\right]\cup\left[(X_1\cap Z_2)\times Y_2\right]. [ ( X 1 × T ) ∪ ( Z 1 × Y 1 ) ] ∩ [ ( X 2 × T ) ∪ ( Z 2 × Y 2 ) ] = [ ( X 1 ∩ X 2 ) × T ] ∪ [ ( Z 1 ∩ X 2 ) × Y 1 ] ∪ [ ( X 1 ∩ Z 2 ) × Y 2 ] . Proof. The assertions are local, and hence can be checked on products of affine open subsets. We may therefore assume that S = Spec A S=\operatorname{Spec} A S = Spec A T = Spec B T=\operatorname{Spec} B T = Spec B I j , K j ⊆ A I_j,K_j\subseteq A I j , K j ⊆ A X j , Z j X_j,Z_j X j , Z j X j ⊆ Z j X_j\subseteq Z_j X j ⊆ Z j K j ⊆ I j K_j\subseteq I_j K j ⊆ I j
Let J ⊆ B J\subseteq B J ⊆ B Y ⊆ T Y\subseteq T Y ⊆ T
I ( X j × T ) = I j ⊗ B and I ( Z j × Y ) = ( K j ⊗ B ) + ( A ⊗ J ) . I(X_j\times T)=I_j\otimes B\qquad\text{and}\qquad I(Z_j\times Y)=(K_j\otimes B)+(A\otimes J). I ( X j × T ) = I j ⊗ B and I ( Z j × Y ) = ( K j ⊗ B ) + ( A ⊗ J ) . Hence we have
I ( ( X j × T ) ∪ ( Z j × Y ) ) = ( I j ⊗ B ) ∩ ( ( K j ⊗ B ) + ( A ⊗ J ) ) = ( K j ⊗ B ) + ( ( I j ⊗ B ) ∩ ( A ⊗ J ) ) . I\left((X_j\times T)\cup(Z_j\times Y)\right)
=(I_j\otimes B)\cap\left((K_j\otimes B)+(A\otimes J)\right)
=(K_j\otimes B)+\left((I_j\otimes B)\cap(A\otimes J)\right). I ( ( X j × T ) ∪ ( Z j × Y ) ) = ( I j ⊗ B ) ∩ ( ( K j ⊗ B ) + ( A ⊗ J ) ) = ( K j ⊗ B ) + ( ( I j ⊗ B ) ∩ ( A ⊗ J ) ) . Using ( I j ⊗ B ) ∩ ( A ⊗ J ) = I j ⊗ J (I_j\otimes B)\cap(A\otimes J)=I_j\otimes J ( I j ⊗ B ) ∩ ( A ⊗ J ) = I j ⊗ J
I ( ( X j × T ) ∪ ( Z j × Y ) ) = ( K j ⊗ B ) + ( I j ⊗ J ) . I\left((X_j\times T)\cup(Z_j\times Y)\right)=(K_j\otimes B)+(I_j\otimes J). I ( ( X j × T ) ∪ ( Z j × Y ) ) = ( K j ⊗ B ) + ( I j ⊗ J ) . Consequently, the ideal of the left-hand side is
( ( K 1 ⊗ B ) + ( I 1 ⊗ J ) ) + ( ( K 2 ⊗ B ) + ( I 2 ⊗ J ) ) = ( K 1 + K 2 ) ⊗ B + ( I 1 + I 2 ) ⊗ J . \left((K_1\otimes B)+(I_1\otimes J)\right)+\left((K_2\otimes B)+(I_2\otimes J)\right)
=(K_1+K_2)\otimes B+(I_1+I_2)\otimes J. ( ( K 1 ⊗ B ) + ( I 1 ⊗ J ) ) + ( ( K 2 ⊗ B ) + ( I 2 ⊗ J ) ) = ( K 1 + K 2 ) ⊗ B + ( I 1 + I 2 ) ⊗ J . On the other hand, the ideal of ( X 1 ∩ X 2 ) × T (X_1\cap X_2)\times T ( X 1 ∩ X 2 ) × T ( I 1 + I 2 ) ⊗ B (I_1+I_2)\otimes B ( I 1 + I 2 ) ⊗ B ( Z 1 ∩ Z 2 ) × Y (Z_1\cap Z_2)\times Y ( Z 1 ∩ Z 2 ) × Y ( ( K 1 + K 2 ) ⊗ B ) + ( A ⊗ J ) ((K_1+K_2)\otimes B)+(A\otimes J) (( K 1 + K 2 ) ⊗ B ) + ( A ⊗ J )
( I 1 + I 2 ) ⊗ B ∩ ( ( ( K 1 + K 2 ) ⊗ B ) + ( A ⊗ J ) ) . (I_1+I_2)\otimes B\cap\left(((K_1+K_2)\otimes B)+(A\otimes J)\right). ( I 1 + I 2 ) ⊗ B ∩ ( (( K 1 + K 2 ) ⊗ B ) + ( A ⊗ J ) ) . Since K 1 + K 2 ⊆ I 1 + I 2 K_1+K_2\subseteq I_1+I_2 K 1 + K 2 ⊆ I 1 + I 2
( I 1 + I 2 ) ⊗ B ∩ ( ( ( K 1 + K 2 ) ⊗ B ) + ( A ⊗ J ) ) = ( ( K 1 + K 2 ) ⊗ B ) + ( ( ( I 1 + I 2 ) ⊗ B ) ∩ ( A ⊗ J ) ) = ( ( K 1 + K 2 ) ⊗ B ) + ( ( I 1 + I 2 ) ⊗ J ) . \begin{aligned}
&(I_1+I_2)\otimes B\cap\left(((K_1+K_2)\otimes B)+(A\otimes J)\right)\\
&\qquad=((K_1+K_2)\otimes B)+\left(((I_1+I_2)\otimes B)\cap(A\otimes J)\right)\\
&\qquad=((K_1+K_2)\otimes B)+((I_1+I_2)\otimes J).
\end{aligned} ( I 1 + I 2 ) ⊗ B ∩ ( (( K 1 + K 2 ) ⊗ B ) + ( A ⊗ J ) ) = (( K 1 + K 2 ) ⊗ B ) + ( (( I 1 + I 2 ) ⊗ B ) ∩ ( A ⊗ J ) ) = (( K 1 + K 2 ) ⊗ B ) + (( I 1 + I 2 ) ⊗ J ) . This is the ideal of the left-hand side, proving the first assertion. For the second assertion, the open subsets T ∖ ∣ Y 1 ∣ T \setminus\lvert Y_1\rvert T ∖ ∣ Y 1 ∣ T ∖ ∣ Y 2 ∣ T \setminus\lvert Y_2\rvert T ∖ ∣ Y 2 ∣ T T T
Let V = Spec ( C ) ⊆ T ∖ ∣ Y 2 ∣ V = \operatorname{Spec}(C) \subseteq T \setminus\lvert Y_2\rvert V = Spec ( C ) ⊆ T ∖ ∣ Y 2 ∣ J 1 , V ⊆ C J_{1,V} \subseteq C J 1 , V ⊆ C Y 1 ∩ V Y_1 \cap V Y 1 ∩ V Y 2 ∩ V = ∅ Y_2 \cap V = \varnothing Y 2 ∩ V = ∅ S × V S \times V S × V
[ ( X 1 × V ) ∪ ( Z 1 × ( Y 1 ∩ V ) ) ] ∩ ( X 2 × V ) . [(X_1 \times V) \cup(Z_1 \times(Y_1 \cap V))] \cap(X_2 \times V). [( X 1 × V ) ∪ ( Z 1 × ( Y 1 ∩ V ))] ∩ ( X 2 × V ) . Its ideal in A ⊗ C A \otimes C A ⊗ C
( ( K 1 ⊗ C ) + ( I 1 ⊗ J 1 , V ) ) + ( I 2 ⊗ C ) = ( ( K 1 + I 2 ) ⊗ C ) + ( I 1 ⊗ J 1 , V ) . ((K_1 \otimes C) + (I_1 \otimes J_{1,V})) + (I_2 \otimes C) = ((K_1 + I_2) \otimes C) + (I_1 \otimes J_{1,V}). (( K 1 ⊗ C ) + ( I 1 ⊗ J 1 , V )) + ( I 2 ⊗ C ) = (( K 1 + I 2 ) ⊗ C ) + ( I 1 ⊗ J 1 , V ) . Over S × V S \times V S × V
[ ( X 1 ∩ X 2 ) × V ] ∪ [ ( Z 1 ∩ X 2 ) × ( Y 1 ∩ V ) ] . [(X_1 \cap X_2) \times V] \cup[(Z_1 \cap X_2) \times(Y_1 \cap V)]. [( X 1 ∩ X 2 ) × V ] ∪ [( Z 1 ∩ X 2 ) × ( Y 1 ∩ V )] . Its ideal is
( ( I 1 + I 2 ) ⊗ C ) ∩ ( ( ( K 1 + I 2 ) ⊗ C ) + ( A ⊗ J 1 , V ) ) . ((I_1 + I_2) \otimes C) \cap(((K_1 + I_2) \otimes C) + (A \otimes J_{1,V})). (( I 1 + I 2 ) ⊗ C ) ∩ ((( K 1 + I 2 ) ⊗ C ) + ( A ⊗ J 1 , V )) . Since K 1 + I 2 ⊆ I 1 + I 2 K_1 + I_2 \subseteq I_1 + I_2 K 1 + I 2 ⊆ I 1 + I 2
( ( I 1 + I 2 ) ⊗ C ) ∩ ( ( ( K 1 + I 2 ) ⊗ C ) + ( A ⊗ J 1 , V ) ) = ( ( K 1 + I 2 ) ⊗ C ) + ( ( I 1 + I 2 ) ⊗ J 1 , V ) = ( ( K 1 + I 2 ) ⊗ C ) + ( I 1 ⊗ J 1 , V ) , \begin{aligned}
((I_1 + I_2) \otimes C) \cap(((K_1 + I_2) \otimes C) + (A \otimes J_{1,V}))
&= ((K_1 + I_2) \otimes C) + ((I_1 + I_2) \otimes J_{1,V}) \\
&= ((K_1 + I_2) \otimes C) + (I_1 \otimes J_{1,V}),
\end{aligned} (( I 1 + I 2 ) ⊗ C ) ∩ ((( K 1 + I 2 ) ⊗ C ) + ( A ⊗ J 1 , V )) = (( K 1 + I 2 ) ⊗ C ) + (( I 1 + I 2 ) ⊗ J 1 , V ) = (( K 1 + I 2 ) ⊗ C ) + ( I 1 ⊗ J 1 , V ) , where the last equality follows from I 2 ⊗ J 1 , V ⊆ I 2 ⊗ C I_2 \otimes J_{1,V} \subseteq I_2 \otimes C I 2 ⊗ J 1 , V ⊆ I 2 ⊗ C T ∖ ∣ Y 2 ∣ T \setminus\lvert Y_2\rvert T ∖ ∣ Y 2 ∣ T ∖ ∣ Y 1 ∣ T \setminus\lvert Y_1\rvert T ∖ ∣ Y 1 ∣ T T T
The previous two lemmas will be used extensively in the next proposition, which proves reducedness of the pointed Hurwitz–Brill–Noether loci via reducedness of the central fiber (and hence scheme-theoretic regeneration).
Proposition 3.8. Let [ π : C → P 1 , p ] [\pi: C \to\mathbb{P}^1,p] [ π : C → P 1 , p ] H g , k t o t H^{\mathrm{tot}}_{g,k} H g , k tot e ⃗ \vec e e k k k
{ L ∈ Pic d ( C ) ∣ h 0 ( C , L − ( m k + i ) p ) ≥ h 0 ( P 1 , O ( σ e ⃗ ) ( − m ) ) for m ≥ 0 , and 0 ≤ i ≤ k − 1 } \{L \in\operatorname{Pic}^{d}(C) \mid h^0(C,L-(mk+i)p) \ge h^0(\mathbb{P}^1,\mathcal{O}(\sigma^{\vec e})(-m)) \text{ for } m \ge0,\text{ and } 0 \le i \le k-1\} { L ∈ Pic d ( C ) ∣ h 0 ( C , L − ( mk + i ) p ) ≥ h 0 ( P 1 , O ( σ e ) ( − m )) for m ≥ 0 , and 0 ≤ i ≤ k − 1 } is reduced, with all components of dimension g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec e)-\operatorname{inv}(\vec e) g − u ( e ) − inv ( e )
Proof. We will prove this by induction on g g g 1 1 1 [15 Lemma 7.3] . We assume the statement is true up to genus g − 1 g-1 g − 1 g g g
Since reducedness is an open condition, we will consider the 1 1 1
ρ : C × B Pic d ( C / B ) ⟶ Pic d ( C / B ) \rho: C \times_B \operatorname{Pic}^{d}(C/B) \longrightarrow\operatorname{Pic}^{d}(C/B) ρ : C × B Pic d ( C / B ) ⟶ Pic d ( C / B ) be the projection, and let L \mathcal{L} L D ⊆ C D \subseteq C D ⊆ C C 0 = X ∪ q E C_0 = X \cup_q E C 0 = X ∪ q E D ∣ X D|_X D ∣ X D ∣ E D|_E D ∣ E m m m D D D
R 1 ρ ∗ ( L ( − ( m k + i ) p ) ( D ) ) = 0 R^1\rho_*\bigl(\mathcal{L}(-(mk+i)p)(D)\bigr)=0 R 1 ρ ∗ ( L ( − ( mk + i ) p ) ( D ) ) = 0 for every relevant pair ( i , m ) (i,m) ( i , m ) D D D p p p D D D p p p C C C C × B Pic d ( C / B ) C \times_B \operatorname{Pic}^{d}(C/B) C × B Pic d ( C / B )
The restriction sequence induces a morphism of vector bundles
φ i , m : ρ ∗ ( L ( − ( m k + i ) p ) ( D ) ) ⟶ ρ ∗ ( L ( − ( m k + i ) p ) ( D ) ∣ D ) . \varphi_{i,m}:\rho_*\bigl(\mathcal{L}(-(mk+i)p)(D)\bigr) \longrightarrow\rho_*\bigl(\mathcal{L}(-(mk+i)p)(D)|_D\bigr). φ i , m : ρ ∗ ( L ( − ( mk + i ) p ) ( D ) ) ⟶ ρ ∗ ( L ( − ( mk + i ) p ) ( D ) ∣ D ) . For a point [ L ] ∈ Pic d ( C / B ) [L] \in\operatorname{Pic}^{d}(C/B) [ L ] ∈ Pic d ( C / B ) b ∈ B b \in B b ∈ B
ker ( φ i , m ∣ [ L ] ) ≅ H 0 ( C b , L ( − ( m k + i ) p b ) ) . \ker\bigl(\varphi_{i,m}|_{[L]}\bigr) \cong H^0\bigl(C_b,L(-(mk+i)p_b)\bigr). ker ( φ i , m ∣ [ L ] ) ≅ H 0 ( C b , L ( − ( mk + i ) p b ) ) . Consequently, the condition
h 0 ( C b , L ( − ( m k + i ) p b ) ) ≥ h 0 ( P 1 , O ( σ i e ⃗ ) ( − m ) ) h^0(C_b,L(-(mk+i)p_b)) \ge h^0(\mathbb{P}^1,\mathcal{O}(\sigma^{i}\vec{e})(-m)) h 0 ( C b , L ( − ( mk + i ) p b )) ≥ h 0 ( P 1 , O ( σ i e ) ( − m )) is equivalent to the rank condition
rk ( φ i , m ∣ [ L ] ) ≤ rk ρ ∗ ( L ( − ( m k + i ) p ) ( D ) ) − h 0 ( P 1 , O ( σ i e ⃗ ) ( − m ) ) . \operatorname{rk}\left(\varphi_{i,m}|_{[L]}\right) \le\operatorname{rk}\rho_*\left(\mathcal{L}(-(mk+i)p)(D)\right)-h^0(\mathbb{P}^1,\mathcal{O}(\sigma^{i}\vec{e})(-m)). rk ( φ i , m ∣ [ L ] ) ≤ rk ρ ∗ ( L ( − ( mk + i ) p ) ( D ) ) − h 0 ( P 1 , O ( σ i e ) ( − m )) . We consider the intersection of all rank conditions above, for each relevant pair ( i , m ) (i,m) ( i , m )
( O X ( q ) ⊗ W ‾ e ⃗ ( π ∣ X , q ) ) × Pic 0 ( E ) ∪ ⋃ t i ( e ⃗ ) ≠ e ⃗ ( O X ( q ) ⊗ W ‾ t i ( e ⃗ ) ( π ∣ X , q ) ) × { O E ( i ⋅ p − i ⋅ q ) } . \left(\mathcal{O}_X(q)\otimes\overline{W}^{\vec{e}}(\pi|_X,q)\right)\times\operatorname{Pic}^0(E)\ \cup\ \bigcup_{t_i(\vec{e})\ne\vec{e}}\left(\mathcal{O}_X(q)\otimes\overline{W}^{t_i(\vec{e})}(\pi|_X,q)\right)\times\left\{\mathcal{O}_E(i\cdot p-i\cdot q)\right\}. ( O X ( q ) ⊗ W e ( π ∣ X , q ) ) × Pic 0 ( E ) ∪ t i ( e ) = e ⋃ ( O X ( q ) ⊗ W t i ( e ) ( π ∣ X , q ) ) × { O E ( i ⋅ p − i ⋅ q ) } . where the union is over 0 ≤ i ≤ k − 1 0\le i\le k-1 0 ≤ i ≤ k − 1 t i ( e ⃗ ) ≠ e ⃗ t_i(\vec{e})\ne\vec{e} t i ( e ) = e
For simplicity of notation, we define
A ( d E , x ) : = { L X ∈ Pic g + r ( X ) ∣ h 0 ( X , L X ( − d E ⋅ q ) ) ≥ x } , A(d_E,x):=\{L_X\in\operatorname{Pic}^{g+r}(X)\mid h^0(X,L_X(-d_E\cdot q))\ge x\}, A ( d E , x ) := { L X ∈ Pic g + r ( X ) ∣ h 0 ( X , L X ( − d E ⋅ q )) ≥ x } , B ( d E , y ) : = { L E ∈ Pic 0 ( E ) ∣ h 0 ( E , L E ( d E ⋅ q − ( m k + i ) ⋅ p ) ) ≥ y } , B(d_E,y):=\{L_E\in\operatorname{Pic}^0(E)\mid h^0(E,L_E(d_E\cdot q-(mk+i)\cdot p))\ge y\}, B ( d E , y ) := { L E ∈ Pic 0 ( E ) ∣ h 0 ( E , L E ( d E ⋅ q − ( mk + i ) ⋅ p )) ≥ y } , and moreover we denote by r i , m : = h 0 ( P 1 , O ( σ i e ⃗ ) ( − m ) ) r_{i,m}:=h^0(\mathbb{P}^1,\mathcal{O}(\sigma^{i}\vec{e})(-m)) r i , m := h 0 ( P 1 , O ( σ i e ) ( − m )) Lemma 3.6 , as well as [15 Lemma 7.1] , the rank condition for a relevant pair ( i , m ) (i,m) ( i , m )
⋃ x + y = r i , m { L X ∣ h 0 ( X , L X ( − q ) ) ≥ x } × { L E ∣ h 0 ( E , L E ( − ( m k + i ) ⋅ p − q ) ) ≥ y } \bigcup_{x+y=r_{i,m}}\{L_X\mid h^0(X,L_X(-q))\ge x\}\times\{L_E\mid h^0(E,L_E(-(mk+i)\cdot p-q))\ge y\} x + y = r i , m ⋃ { L X ∣ h 0 ( X , L X ( − q )) ≥ x } × { L E ∣ h 0 ( E , L E ( − ( mk + i ) ⋅ p − q )) ≥ y } ∪ ⋃ x + y = r i , m + 1 { L X ∣ h 0 ( X , L X ) ≥ x } × { L E ∣ h 0 ( E , L E ( − ( m k + i ) ⋅ p ) ) ≥ y } \cup\bigcup_{x+y=r_{i,m}+1}\{L_X\mid h^0(X,L_X)\ge x\}\times\{L_E\mid h^0(E,L_E(-(mk+i)\cdot p))\ge y\} ∪ x + y = r i , m + 1 ⋃ { L X ∣ h 0 ( X , L X ) ≥ x } × { L E ∣ h 0 ( E , L E ( − ( mk + i ) ⋅ p )) ≥ y } in the central fiber Pic d X ( X ) × Pic d E ( E ) \operatorname{Pic}^{d_X}(X)\times\operatorname{Pic}^{d_E}(E) Pic d X ( X ) × Pic d E ( E ) d X + d E = g + r d_X+d_E=g+r d X + d E = g + r Pic g + r ( X ) × Pic 0 ( E ) \operatorname{Pic}^{g+r}(X)\times\operatorname{Pic}^0(E) Pic g + r ( X ) × Pic 0 ( E )
⋃ x + y = r i , m A ( d E + 1 , x ) × B ( d E − 1 , y ) ∪ ⋃ x + y = r i , m + 1 A ( d E , x ) × B ( d E , y ) . \bigcup_{x+y=r_{i,m}}A(d_E+1,x)\times B(d_E-1,y)\ \cup\ \bigcup_{x+y=r_{i,m}+1}A(d_E,x)\times B(d_E,y). x + y = r i , m ⋃ A ( d E + 1 , x ) × B ( d E − 1 , y ) ∪ x + y = r i , m + 1 ⋃ A ( d E , x ) × B ( d E , y ) . Next, we remark that, as we vary d E d_E d E Z \mathbb{Z} Z
First, if the degree d E − 1 − m k − i d_E-1-mk-i d E − 1 − mk − i
⋃ x + y = r i , m A ( d E + 1 , x ) × B ( d E − 1 , y ) = A ( d E + 1 , r i , m ) × Pic 0 ( E ) . \bigcup_{x+y=r_{i,m}}A(d_E+1,x)\times B(d_E-1,y)=A(d_E+1,r_{i,m})\times\operatorname{Pic}^0(E). x + y = r i , m ⋃ A ( d E + 1 , x ) × B ( d E − 1 , y ) = A ( d E + 1 , r i , m ) × Pic 0 ( E ) . Indeed, if y > 0 y>0 y > 0 B ( d E − 1 , y ) = ∅ B(d_E-1,y)=\varnothing B ( d E − 1 , y ) = ∅ y < 0 y<0 y < 0
A ( d E + 1 , x + 1 ) × Pic 0 ( E ) ⊆ A ( d E + 1 , x ) × Pic 0 ( E ) . A(d_E+1,x+1)\times\operatorname{Pic}^0(E)\subseteq A(d_E+1,x)\times\operatorname{Pic}^0(E). A ( d E + 1 , x + 1 ) × Pic 0 ( E ) ⊆ A ( d E + 1 , x ) × Pic 0 ( E ) . Similarly, when d E − 1 − m k − i ≥ 1 d_E-1-mk-i\ge1 d E − 1 − mk − i ≥ 1
h 0 ( E , L E ( ( d E − 1 ) ⋅ q − ( m k + i ) ⋅ p ) ) = d E − 1 − m k − i . h^0(E,L_E((d_E-1)\cdot q-(mk+i)\cdot p))=d_E-1-mk-i. h 0 ( E , L E (( d E − 1 ) ⋅ q − ( mk + i ) ⋅ p )) = d E − 1 − mk − i . A similar argument as above gives in this case:
⋃ x + y = r i , m A ( d E + 1 , x ) × B ( d E − 1 , y ) = A ( d E + 1 , r i , m + 1 + m k + i − d E ) × Pic 0 ( E ) . \bigcup_{x+y=r_{i,m}}A(d_E+1,x)\times B(d_E-1,y)=A(d_E+1,r_{i,m}+1+mk+i-d_E)\times\operatorname{Pic}^0(E). x + y = r i , m ⋃ A ( d E + 1 , x ) × B ( d E − 1 , y ) = A ( d E + 1 , r i , m + 1 + mk + i − d E ) × Pic 0 ( E ) . Lastly, when d E − 1 − m k − i = 0 d_E-1-mk-i=0 d E − 1 − mk − i = 0 E E E h 0 ( E , − ) h^0(E,-) h 0 ( E , − ) y = 0 y=0 y = 0 y = 1 y=1 y = 1
⋃ x + y = r i , m A ( d E + 1 , x ) × B ( d E − 1 , y ) = A ( d E + 1 , r i , m ) × Pic 0 ( E ) ∪ A ( d E + 1 , r i , m − 1 ) × { O E ( i ⋅ p − i ⋅ q ) } . \bigcup_{x+y=r_{i,m}} A(d_E+1,x)\times B(d_E-1,y)=A(d_E+1,r_{i,m})\times\operatorname{Pic}^{0}(E)\cup A(d_E+1,r_{i,m}-1)\times\{\mathcal{O}_E(i\cdot p-i\cdot q)\}. x + y = r i , m ⋃ A ( d E + 1 , x ) × B ( d E − 1 , y ) = A ( d E + 1 , r i , m ) × Pic 0 ( E ) ∪ A ( d E + 1 , r i , m − 1 ) × { O E ( i ⋅ p − i ⋅ q )} . A similar argument works for the union
⋃ x + y = r i , m + 1 A ( d E , x ) × B ( d E , y ) , \bigcup_{x+y=r_{i,m+1}} A(d_E,x)\times B(d_E,y), x + y = r i , m + 1 ⋃ A ( d E , x ) × B ( d E , y ) , which is scheme theoretically equal to
A ( d E , r i , m + 1 ) × Pic 0 ( E ) , if d E − m k − i < 0 ; A(d_E,r_{i,m}+1)\times\operatorname{Pic}^{0}(E),\quad\text{if }d_E-mk-i<0; A ( d E , r i , m + 1 ) × Pic 0 ( E ) , if d E − mk − i < 0 ; A ( d E , r i , m + 1 + m k + i − d E ) × Pic 0 ( E ) , if d E − m k − i ≥ 1 ; A(d_E,r_{i,m}+1+mk+i-d_E)\times\operatorname{Pic}^{0}(E),\quad\text{if }d_E-mk-i\geq1; A ( d E , r i , m + 1 + mk + i − d E ) × Pic 0 ( E ) , if d E − mk − i ≥ 1 ; or
A ( d E , r i , m + 1 ) × Pic 0 ( E ) ∪ A ( d E , r i , m ) × { O E ( i ⋅ p − i ⋅ q ) } , if d E − m k − i = 0. A(d_E,r_{i,m}+1)\times\operatorname{Pic}^{0}(E)\cup A(d_E,r_{i,m})\times\{\mathcal{O}_E(i\cdot p-i\cdot q)\},\quad\text{if }d_E-mk-i=0. A ( d E , r i , m + 1 ) × Pic 0 ( E ) ∪ A ( d E , r i , m ) × { O E ( i ⋅ p − i ⋅ q )} , if d E − mk − i = 0. Indeed, we compute the union as d E d_E d E Z \mathbb{Z} Z
If d E < m k + i d_E<mk+i d E < mk + i
A ( d E + 1 , r i , m ) × Pic 0 ( E ) ∪ A ( d E , r i , m + 1 ) × Pic 0 ( E ) = A ( d E + 1 , r i , m ) × Pic 0 ( E ) . A(d_E+1,r_{i,m})\times\operatorname{Pic}^{0}(E)\cup A(d_E,r_{i,m}+1)\times\operatorname{Pic}^{0}(E)=A(d_E+1,r_{i,m})\times\operatorname{Pic}^{0}(E). A ( d E + 1 , r i , m ) × Pic 0 ( E ) ∪ A ( d E , r i , m + 1 ) × Pic 0 ( E ) = A ( d E + 1 , r i , m ) × Pic 0 ( E ) . If d E = m k + i d_E=mk+i d E = mk + i
A ( d E + 1 , r i , m ) × Pic 0 ( E ) ∪ A ( d E , r i , m + 1 ) × Pic 0 ( E ) ∪ A ( d E , r i , m ) × { O E ( i ⋅ p − i ⋅ q ) } . A(d_E+1,r_{i,m})\times\operatorname{Pic}^{0}(E)\cup A(d_E,r_{i,m}+1)\times\operatorname{Pic}^{0}(E)\cup A(d_E,r_{i,m})\times\{\mathcal{O}_E(i\cdot p-i\cdot q)\}. A ( d E + 1 , r i , m ) × Pic 0 ( E ) ∪ A ( d E , r i , m + 1 ) × Pic 0 ( E ) ∪ A ( d E , r i , m ) × { O E ( i ⋅ p − i ⋅ q )} . As before, the second term is contained scheme theoretically in the first term, and hence the union is
A ( d E + 1 , r i , m ) × Pic 0 ( E ) ∪ A ( d E , r i , m ) × { O E ( i ⋅ p − i ⋅ q ) } . A(d_E+1,r_{i,m})\times\operatorname{Pic}^{0}(E)\cup A(d_E,r_{i,m})\times\{\mathcal{O}_E(i\cdot p-i\cdot q)\}. A ( d E + 1 , r i , m ) × Pic 0 ( E ) ∪ A ( d E , r i , m ) × { O E ( i ⋅ p − i ⋅ q )} . If d E = m k + i + 1 d_E=mk+i+1 d E = mk + i + 1
A ( d E + 1 , r i , m ) × Pic 0 ( E ) ∪ A ( d E + 1 , r i , m − 1 ) × { O E ( i ⋅ p − i ⋅ q ) } ∪ A ( d E , r i , m ) × Pic 0 ( E ) . A(d_E+1,r_{i,m})\times\operatorname{Pic}^{0}(E)\cup A(d_E+1,r_{i,m}-1)\times\{\mathcal{O}_E(i\cdot p-i\cdot q)\}\cup A(d_E,r_{i,m})\times\operatorname{Pic}^{0}(E). A ( d E + 1 , r i , m ) × Pic 0 ( E ) ∪ A ( d E + 1 , r i , m − 1 ) × { O E ( i ⋅ p − i ⋅ q )} ∪ A ( d E , r i , m ) × Pic 0 ( E ) . The first term is contained in the third and hence the union is
A ( d E + 1 , r i , m − 1 ) × { O E ( i ⋅ p − i ⋅ q ) } ∪ A ( d E , r i , m ) × Pic 0 ( E ) . A(d_E+1,r_{i,m}-1)\times\{\mathcal{O}_E(i\cdot p-i\cdot q)\}\cup A(d_E,r_{i,m})\times\operatorname{Pic}^{0}(E). A ( d E + 1 , r i , m − 1 ) × { O E ( i ⋅ p − i ⋅ q )} ∪ A ( d E , r i , m ) × Pic 0 ( E ) . If d E > m k + i + 1 d_E>mk+i+1 d E > mk + i + 1
A ( d E + 1 , r i , m + 1 + m k + i − d E ) × Pic 0 ( E ) ∪ A ( d E , r i , m + 1 + m k + i − d E ) × Pic 0 ( E ) . A(d_E+1,r_{i,m}+1+mk+i-d_E)\times\operatorname{Pic}^{0}(E)\cup A(d_E,r_{i,m}+1+mk+i-d_E)\times\operatorname{Pic}^{0}(E). A ( d E + 1 , r i , m + 1 + mk + i − d E ) × Pic 0 ( E ) ∪ A ( d E , r i , m + 1 + mk + i − d E ) × Pic 0 ( E ) . The first term is contained in the second and hence this is just
A ( d E , r i , m + 1 + m k + i − d E ) × Pic 0 ( E ) . A(d_E,r_{i,m}+1+mk+i-d_E)\times\operatorname{Pic}^{0}(E). A ( d E , r i , m + 1 + mk + i − d E ) × Pic 0 ( E ) . We are now ready to consider the intersection of these rank degeneracy condition as the degree d E d_E d E Z \mathbb{Z} Z
for all d E < m k + 1 , we have A ( m k + i , r i , m ) ⊆ A ( d E + 1 , r i , m ) , and \text{for all }d_E<mk+1,\text{ we have }A(mk+i,r_{i,m})\subseteq A(d_E+1,r_{i,m}),\text{ and} for all d E < mk + 1 , we have A ( mk + i , r i , m ) ⊆ A ( d E + 1 , r i , m ) , and for all d E > m k + i + 1 , we have A ( m k + i + 2 , r i , m − 1 ) ⊆ A ( d E , r i , m + 1 + m k + i − d E ) . \text{for all }d_E>mk+i+1,\text{ we have }A(mk+i+2,r_{i,m}-1)\subseteq A(d_E,r_{i,m}+1+mk+i-d_E). for all d E > mk + i + 1 , we have A ( mk + i + 2 , r i , m − 1 ) ⊆ A ( d E , r i , m + 1 + mk + i − d E ) . In particular, the intersection of these rank conditions as d E d_E d E Z \mathbb{Z} Z d E d_E d E m k + i − 1 mk+i-1 mk + i − 1 m k + i + 2 mk+i+2 mk + i + 2
Furthermore, we have the scheme-theoretic inclusions
A ( m k + i + 1 , r i , m ) ⊆ A ( m k + i , r i , m ) A(mk+i+1,r_{i,m})\subseteq A(mk+i,r_{i,m}) A ( mk + i + 1 , r i , m ) ⊆ A ( mk + i , r i , m ) and
A ( m k + i + 1 , r i , m ) ⊆ A ( m k + i + 2 , r i , m − 1 ) . A(mk+i+1,r_{i,m})\subseteq A(mk+i+2,r_{i,m}-1). A ( mk + i + 1 , r i , m ) ⊆ A ( mk + i + 2 , r i , m − 1 ) . Consequently, the first and fourth loci are redundant and it is enough to consider
d E ∈ { m k + i , m k + i + 1 } d_E\in\{mk+i,mk+i+1\} d E ∈ { mk + i , mk + i + 1 } in the intersection.
Using the general fact of scheme-theoretic distributivity of intersection across unions in Lemma 3.7 , we obtain
⋂ d E ∈ Z [ ⋃ x + y = r i , m A ( d E + 1 , x ) × B ( d E − 1 , y ) ∪ ⋃ x + y = r i , m + 1 A ( d E , x ) × B ( d E , y ) ] = ( A ( m k + i + 1 , r i , m ) × Pic 0 ( E ) ) ∪ ( A ( m k + i , r i , m ) ∩ A ( m k + i + 2 , r i , m − 1 ) ) × { O E ( i ( p − q ) ) } . \begin{aligned}
\bigcap_{d_E\in\mathbb{Z}}\left[\bigcup_{x+y=r_{i,m}} A(d_E+1,x)\times B(d_E-1,y)\ \cup\ \bigcup_{x+y=r_{i,m+1}} A(d_E,x)\times B(d_E,y)\right] \\
&=\left(A(mk+i+1,r_{i,m})\times\operatorname{Pic}^{0}(E)\right)\cup\left(A(mk+i,r_{i,m})\cap A(mk+i+2,r_{i,m}-1)\right)\times\{O_E(i(p-q))\}.
\end{aligned} d E ∈ Z ⋂ x + y = r i , m ⋃ A ( d E + 1 , x ) × B ( d E − 1 , y ) ∪ x + y = r i , m + 1 ⋃ A ( d E , x ) × B ( d E , y ) = ( A ( mk + i + 1 , r i , m ) × Pic 0 ( E ) ) ∪ ( A ( mk + i , r i , m ) ∩ A ( mk + i + 2 , r i , m − 1 ) ) × { O E ( i ( p − q ))} . We now intersect over all relevant pairs ( i , m ) (i,m) ( i , m ) p − q p-q p − q k k k
O E ( i ( p − q ) ) , 0 ≤ i ≤ k − 1 , O_E(i(p-q)),\qquad0\leq i\leq k-1, O E ( i ( p − q )) , 0 ≤ i ≤ k − 1 , are pairwise distinct. Therefore, a repeated application of Lemma 3.7 gives
⋂ 0 ≤ i ≤ k − 1 m ≥ 0 ⋂ d E ∈ Z [ ⋃ x + y = r i , m A ( d E + 1 , x ) × B ( d E − 1 , y ) ∪ ⋃ x + y = r i , m + 1 A ( d E , x ) × B ( d E , y ) ] = ( ⋂ 0 ≤ i ≤ k − 1 m ≥ 0 A ( m k + i + 1 , r i , m ) ) × Pic 0 ( E ) ∪ \begin{aligned}
\bigcap_{\substack{0\leq i\leq k-1\\m\geq0}}\bigcap_{d_E\in\mathbb{Z}}\left[\bigcup_{x+y=r_{i,m}} A(d_E+1,x)\times B(d_E-1,y)\ \cup\ \bigcup_{x+y=r_{i,m+1}} A(d_E,x)\times B(d_E,y)\right] \\
&=\left(\bigcap_{\substack{0\leq i\leq k-1\\m\geq0}}A(mk+i+1,r_{i,m})\right)\times\operatorname{Pic}^{0}(E)\ \cup
\end{aligned} 0 ≤ i ≤ k − 1 m ≥ 0 ⋂ d E ∈ Z ⋂ x + y = r i , m ⋃ A ( d E + 1 , x ) × B ( d E − 1 , y ) ∪ x + y = r i , m + 1 ⋃ A ( d E , x ) × B ( d E , y ) = 0 ≤ i ≤ k − 1 m ≥ 0 ⋂ A ( mk + i + 1 , r i , m ) × Pic 0 ( E ) ∪ ⋃ i = 0 k − 1 [ ( ⋂ m ≥ 0 ( A ( m k + i , r i , m ) ∩ A ( m k + i + 2 , r i , m − 1 ) ) ) ∩ ( ⋂ m ≥ 0 0 ≤ j ≤ k − 1 j ≠ i A ( m k + j + 1 , r j , m ) ) ] × { O E ( i ( p − q ) ) } . \bigcup_{i=0}^{k-1}\left[\left(\bigcap_{m\geq0}\left(A(mk+i,r_{i,m})\cap A(mk+i+2,r_{i,m}-1)\right)\right)\cap\left(\bigcap_{\substack{m\geq0\\0\leq j\leq k-1\\j\neq i}}A(mk+j+1,r_{j,m})\right)\right]\times\{O_E(i(p-q))\}. i = 0 ⋃ k − 1 ( m ≥ 0 ⋂ ( A ( mk + i , r i , m ) ∩ A ( mk + i + 2 , r i , m − 1 ) ) ) ∩ m ≥ 0 0 ≤ j ≤ k − 1 j = i ⋂ A ( mk + j + 1 , r j , m ) × { O E ( i ( p − q ))} . If we replace L X L_X L X L X ( − q ) L_X(-q) L X ( − q ) g − 1 g-1 g − 1 e ⃗ \vec e e
h 0 ( X , L X − ( m k + i + 1 ) q ) = h 0 ( X , L X ( − q ) − ( m k + i ) q ) , h^0(X,L_X-(mk+i+1)q)=h^0(X,L_X(-q)-(mk+i)q), h 0 ( X , L X − ( mk + i + 1 ) q ) = h 0 ( X , L X ( − q ) − ( mk + i ) q ) , and by the induction hypothesis
⋂ 0 ≤ i ≤ k − 1 m ≥ 0 A ( m k + i + 1 , r i , m ) = O X ( q ) ⊗ W ‾ e ⃗ ( π X : X → P X , q ) , \bigcap_{\substack{0\leq i\leq k-1\\m\geq0}}A(mk+i+1,r_{i,m})=O_X(q)\otimes\overline{W}^{\vec e}(\pi_X:X\to P_X,q), 0 ≤ i ≤ k − 1 m ≥ 0 ⋂ A ( mk + i + 1 , r i , m ) = O X ( q ) ⊗ W e ( π X : X → P X , q ) , with the reduced scheme structure.
For each fixed i i i
O X ( q ) ⊗ W ‾ t i ( e ⃗ ) ( π X , q ) . O_X(q)\otimes\overline{W}^{t_i(\vec e)}(\pi_X,q). O X ( q ) ⊗ W t i ( e ) ( π X , q ) . To identify its defining conditions, put M = L X ( − q ) M=L_X(-q) M = L X ( − q )
h 0 ( X , M ( − ( m k + j ) q ) ) ≥ r j , m for j ≠ i , h^0(X,M(-(mk+j)q))\geq r_{j,m}\quad\text{for }j\neq i, h 0 ( X , M ( − ( mk + j ) q )) ≥ r j , m for j = i , together with
h 0 ( X , M ( − ( m k + i − 1 ) q ) ) ≥ r i , m , h^0(X,M(-(mk+i-1)q))\geq r_{i,m}, h 0 ( X , M ( − ( mk + i − 1 ) q )) ≥ r i , m , and
h 0 ( X , M ( − ( m k + i + 1 ) q ) ) ≥ r i , m − 1. h^0(X,M(-(mk+i+1)q))\geq r_{i,m}-1. h 0 ( X , M ( − ( mk + i + 1 ) q )) ≥ r i , m − 1. We first explain which conditions at the omitted twists m k + i mk+i mk + i
r − 1 , m = r k − 1 , m − 1 , r k , m = r 0 , m + 1 . r_{-1,m}=r_{k-1,m-1},\qquad r_{k,m}=r_{0,m+1}. r − 1 , m = r k − 1 , m − 1 , r k , m = r 0 , m + 1 . Fix m m m m k + i > 0 mk+i>0 mk + i > 0 r i − 1 , m r_{i-1,m} r i − 1 , m r i , m r_{i,m} r i , m
h 0 ( X , M ( − ( m k + i − 1 ) q ) ) ≥ r i , m and h 0 ( X , M ( − ( m k + i − 1 ) q ) ) ≥ r i − 1 , m . h^0(X,M(-(mk+i-1)q))\geq r_{i,m}\quad\text{and}\quad h^0(X,M(-(mk+i-1)q))\geq r_{i-1,m}. h 0 ( X , M ( − ( mk + i − 1 ) q )) ≥ r i , m and h 0 ( X , M ( − ( mk + i − 1 ) q )) ≥ r i − 1 , m . At the following twist, it imposes the conditions
h 0 ( X , M ( − ( m k + i + 1 ) q ) ) ≥ r i , m − 1 and h 0 ( X , M ( − ( m k + i + 1 ) q ) ) ≥ r i + 1 , m . h^0(X,M(-(mk+i+1)q)) \ge r_{i,m}-1 \quad\text{and}\quad h^0(X,M(-(mk+i+1)q)) \ge r_{i+1,m}. h 0 ( X , M ( − ( mk + i + 1 ) q )) ≥ r i , m − 1 and h 0 ( X , M ( − ( mk + i + 1 ) q )) ≥ r i + 1 , m . Since
r i − 1 , m ≥ r i , m , r i + 1 , m ≥ r i , m − 1 , r_{i-1,m} \ge r_{i,m}, \qquad r_{i+1,m} \ge r_{i,m}-1, r i − 1 , m ≥ r i , m , r i + 1 , m ≥ r i , m − 1 , the two additional conditions displayed above are redundant.
Moreover, removing one point decreases the dimension of sections by at most one, while adding one point cannot decrease it. Consequently, the conditions at the preceding and following twists imply
h 0 ( X , M ( − ( m k + i ) q ) ) ≥ max { r i − 1 , m − 1 , r i + 1 , m } . h^0(X,M(-(mk+i)q)) \ge\max\{r_{i-1,m}-1,\ r_{i+1,m}\}. h 0 ( X , M ( − ( mk + i ) q )) ≥ max { r i − 1 , m − 1 , r i + 1 , m } . Equivalently, in the original notation, the implicit conditions are
h 0 ( X , L X − ( m k + i + 1 ) q ) ≥ max { r i − 1 , m − 1 , r i + 1 , m } . h^0(X,L_X-(mk+i+1)q) \ge\max\{r_{i-1,m}-1,\ r_{i+1,m}\}. h 0 ( X , L X − ( mk + i + 1 ) q ) ≥ max { r i − 1 , m − 1 , r i + 1 , m } . These implications hold scheme-theoretically for the determinantal loci. Indeed, the restriction matrices for consecutive twists are obtained from one another by adding or deleting one column. The corresponding inclusions of determinantal ideals follow from containment of minors and expansion along this column. Thus adjoining the implicit conditions does not change the scheme structure of the intersection.
We will show that the intersection
[ ( ⋂ m ≥ 0 ( A ( m k + i , r i , m ) ∩ A ( m k + i + 2 , r i , m − 1 ) ) ) ∩ ( ⋂ m ≥ 0 0 ≤ j ≤ k − 1 j ≠ i A ( m k + j + 1 , r j , m ) ) ] \left[
\left(\bigcap_{m\ge0}\left(A(mk+i,r_{i,m})\cap A(mk+i+2,r_{i,m}-1)\right)\right)
\cap
\left(\bigcap_{\substack{m\ge0\\0\le j\le k-1\\j\ne i}} A(mk+j+1,r_{j,m})\right)
\right] ( m ≥ 0 ⋂ ( A ( mk + i , r i , m ) ∩ A ( mk + i + 2 , r i , m − 1 ) ) ) ∩ m ≥ 0 0 ≤ j ≤ k − 1 j = i ⋂ A ( mk + j + 1 , r j , m ) cuts out the locus O X ( q ) ⊗ W t i ( e ⃗ ) ( π X , q ) O_X(q)\otimes W^{t_i(\vec e)}(\pi_X,q) O X ( q ) ⊗ W t i ( e ) ( π X , q ) M M M a ( t i ( e ⃗ ) ) \mathbf{a}(t_i(\vec e)) a ( t i ( e ))
r j , m = # { a ∈ a ( e ⃗ ) ∣ a ≥ m k + j } , r_{j,m}=\#\{a\in\mathbf{a}(\vec e)\mid a\ge mk+j\}, r j , m = # { a ∈ a ( e ) ∣ a ≥ mk + j } , and consequently we obtain
h 0 ( P 1 , O ( σ j t i ( e ⃗ ) ( − m ) ) ) = # { a ∈ a ( t i ( e ⃗ ) ) ∣ a ≥ m k + j } = r j , m when j ≠ i . h^0\left(\mathbb{P}^1,\mathcal{O}(\sigma^{j}t_i(\vec e)(-m))\right)=\#\{a\in\mathbf{a}(t_i(\vec e))\mid a\ge mk+j\}=r_{j,m}\quad\text{when }j\ne i. h 0 ( P 1 , O ( σ j t i ( e ) ( − m )) ) = # { a ∈ a ( t i ( e )) ∣ a ≥ mk + j } = r j , m when j = i . Moreover, we have
r i − 1 , m − r i , m = { 1 , m k + i − 1 ∈ a ( e ⃗ ) , 0 , m k + i − 1 ∉ a ( e ⃗ ) , r_{i-1,m}-r_{i,m}=
\begin{cases}
1, & mk+i-1\in\mathbf{a}(\vec e),\\
0, & mk+i-1\notin\mathbf{a}(\vec e),
\end{cases} r i − 1 , m − r i , m = { 1 , 0 , mk + i − 1 ∈ a ( e ) , mk + i − 1 ∈ / a ( e ) , and
r i , m − r i + 1 , m = { 1 , m k + i ∈ a ( e ⃗ ) , 0 , m k + i ∉ a ( e ⃗ ) . r_{i,m}-r_{i+1,m}=
\begin{cases}
1, & mk+i\in\mathbf{a}(\vec e),\\
0, & mk+i\notin\mathbf{a}(\vec e).
\end{cases} r i , m − r i + 1 , m = { 1 , 0 , mk + i ∈ a ( e ) , mk + i ∈ / a ( e ) . It follows that
max { r i − 1 , m − 1 , r i + 1 , m } = { r i , m − 1 , m k + i ∈ a ( e ⃗ ) and m k + i − 1 ∉ a ( e ⃗ ) , r i , m , otherwise . \max\{r_{i-1,m}-1,\ r_{i+1,m}\}=
\begin{cases}
r_{i,m}-1, & mk+i\in\mathbf{a}(\vec e)\text{ and }mk+i-1\notin\mathbf{a}(\vec e),\\
r_{i,m}, & \text{otherwise}.
\end{cases} max { r i − 1 , m − 1 , r i + 1 , m } = { r i , m − 1 , r i , m , mk + i ∈ a ( e ) and mk + i − 1 ∈ / a ( e ) , otherwise . Thus the bound at m k + i mk+i mk + i m k + i mk+i mk + i a ( e ⃗ ) \mathbf{a}(\vec e) a ( e )
h 0 ( P 1 , O ( σ i t i ( e ⃗ ) ( − m ) ) ) = # { a ∈ a ( t i ( e ⃗ ) ) ∣ a ≥ m k + i } = max { r i − 1 , m − 1 , r i + 1 , m } . h^0\left(\mathbb{P}^1,\mathcal{O}(\sigma^{i}t_i(\vec e)(-m))\right)=\#\{a\in\mathbf{a}(t_i(\vec e))\mid a\ge mk+i\}=\max\{r_{i-1,m}-1,\ r_{i+1,m}\}. h 0 ( P 1 , O ( σ i t i ( e ) ( − m )) ) = # { a ∈ a ( t i ( e )) ∣ a ≥ mk + i } = max { r i − 1 , m − 1 , r i + 1 , m } . We have therefore obtained, scheme-theoretically, exactly the conditions
h 0 ( X , M ( − ( m k + j ) q ) ) ≥ h 0 ( P 1 , O ( σ j t i ( e ⃗ ) ( − m ) ) ) , m ≥ 0 , 0 ≤ j ≤ k − 1. h^0(X,M(-(mk+j)q))\ge h^0\left(\mathbb{P}^1,\mathcal{O}(\sigma^{j}t_i(\vec e)(-m))\right),\qquad m\ge0,\quad0\le j\le k-1. h 0 ( X , M ( − ( mk + j ) q )) ≥ h 0 ( P 1 , O ( σ j t i ( e ) ( − m )) ) , m ≥ 0 , 0 ≤ j ≤ k − 1. In particular, using the induction hypothesis, the locus in Pic g + r ( X ) × Pic 0 ( E ) \operatorname{Pic}^{g+r}(X) \times\operatorname{Pic}^{0}(E) Pic g + r ( X ) × Pic 0 ( E )
O X ( q ) ⊗ W ‾ e ⃗ ( π X , q ) × Pic 0 ( E ) ∪ ⋃ i = 0 k − 1 O X ( q ) ⊗ W ‾ t i ( e ⃗ ) ( π X , q ) × { O E ( i ( p − q ) ) } \mathcal{O}_{X}(q) \otimes\overline{W}^{\vec{e}}(\pi_{X},q) \times\operatorname{Pic}^{0}(E) \cup\bigcup_{i=0}^{k-1} \mathcal{O}_{X}(q) \otimes\overline{W}^{t_{i}(\vec{e})}(\pi_{X},q) \times\{\mathcal{O}_{E}(i(p-q))\} O X ( q ) ⊗ W e ( π X , q ) × Pic 0 ( E ) ∪ i = 0 ⋃ k − 1 O X ( q ) ⊗ W t i ( e ) ( π X , q ) × { O E ( i ( p − q ))} with the reduced scheme structure. This is scheme-theoretically the same as
O X ( q ) ⊗ W ‾ e ⃗ ( π X , q ) × Pic 0 ( E ) ∪ ⋃ t i ( e ⃗ ) ≠ e ⃗ O X ( q ) ⊗ W ‾ t i ( e ⃗ ) ( π X , q ) × { O E ( i ( p − q ) ) } \mathcal{O}_{X}(q) \otimes\overline{W}^{\vec{e}}(\pi_{X},q) \times\operatorname{Pic}^{0}(E) \cup\bigcup_{t_{i}(\vec{e})\ne\vec{e}} \mathcal{O}_{X}(q) \otimes\overline{W}^{t_{i}(\vec{e})}(\pi_{X},q) \times\{\mathcal{O}_{E}(i(p-q))\} O X ( q ) ⊗ W e ( π X , q ) × Pic 0 ( E ) ∪ t i ( e ) = e ⋃ O X ( q ) ⊗ W t i ( e ) ( π X , q ) × { O E ( i ( p − q ))} with the reduced scheme structure. Thus the central fiber is reduced and all its components have dimension g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec{e})-\operatorname{inv}(\vec{e}) g − u ( e ) − inv ( e )
In particular, the central fiber of W e ⃗ ( π , p ) \mathcal{W}^{\vec{e}}(\pi,p) W e ( π , p )
O X ( q ) ⊗ W ‾ e ⃗ ( π X , q ) × Pic 0 ( E ) ∪ ⋃ i = 0 k − 1 O X ( q ) ⊗ W ‾ t i ( e ⃗ ) ( π X , q ) × { O E ( i ( p − q ) ) } \mathcal{O}_{X}(q) \otimes\overline{W}^{\vec{e}}(\pi_{X},q) \times\operatorname{Pic}^{0}(E) \cup\bigcup_{i=0}^{k-1} \mathcal{O}_{X}(q) \otimes\overline{W}^{t_{i}(\vec{e})}(\pi_{X},q) \times\{\mathcal{O}_{E}(i(p-q))\} O X ( q ) ⊗ W e ( π X , q ) × Pic 0 ( E ) ∪ i = 0 ⋃ k − 1 O X ( q ) ⊗ W t i ( e ) ( π X , q ) × { O E ( i ( p − q ))} endowed with the reduced structure. By Theorem 3.5 , this inclusion is an equality on supports. Since the containing scheme is reduced, it is therefore an equality of schemes.
Finally, assume that the rank conditions cut out an additional generic component Δ \Delta Δ Δ 0 \Delta_{0} Δ 0 Δ \Delta Δ W e ⃗ ( π , p ) \mathcal{W}^{\vec{e}}(\pi,p) W e ( π , p ) Δ 0 \Delta_{0} Δ 0 Δ 0 \Delta_{0} Δ 0
Therefore, the generic fiber is reduced and has pure dimension g − u ( e ⃗ ) − inv ( e ⃗ ) g-u(\vec{e})-\operatorname{inv}(\vec{e}) g − u ( e ) − inv ( e )
The proof of Proposition 3.8 also establishes the following scheme-theoretic regeneration statement, which we state separately for its independent interest.
Proposition 3.9 (Scheme-theoretic regeneration). Let C → π P → B \mathcal{C} \xrightarrow{\pi} \mathcal{P} \to B C π P → B C ∗ → P ∗ → B ∗ \mathcal{C}^{*} \to\mathcal{P}^{*} \to B^{*} C ∗ → P ∗ → B ∗ e ⃗ ∈ Z ≥ 0 k \vec{e} \in\mathbb{Z}_{\ge0}^{k} e ∈ Z ≥ 0 k r = k + ∣ e ⃗ ∣ − 1 r=k+|\vec{e}|-1 r = k + ∣ e ∣ − 1 d = g + r d=g+r d = g + r
We consider the schematic closure of W ‾ e ⃗ ( C ∗ → P ∗ ) \overline{\mathcal{W}}^{\vec{e}}(\mathcal{C}^{*}\to\mathcal{P}^{*}) W e ( C ∗ → P ∗ ) W ‾ e ⃗ ( π , p ) \overline{\mathcal{W}}^{\vec{e}}(\pi,p) W e ( π , p ) Pic d ( C / B ) \operatorname{Pic}^{d}(\mathcal{C}/B) Pic d ( C / B ) W ‾ e ⃗ ( π , p ) \overline{\mathcal{W}}^{\vec{e}}(\pi,p) W e ( π , p ) B B B Pic d ( X ) × Pic 0 ( E ) \operatorname{Pic}^{d}(X)\times\operatorname{Pic}^{0}(E) Pic d ( X ) × Pic 0 ( E )
( O X ( q ) ⊗ W ‾ e ⃗ ( π X , q ) ) × Pic 0 ( E ) ∪ ⋃ 0 ≤ i < k t i ( e ⃗ ) ≠ e ⃗ ( O X ( q ) ⊗ W ‾ t i ( e ⃗ ) ( π X , q ) ) × { O E ( i ( p − q ) ) } , \left(\mathcal{O}_{X}(q)\otimes\overline{W}^{\vec{e}}(\pi_{X},q)\right)\times\operatorname{Pic}^{0}(E)\ \cup\ \bigcup_{\substack{0\le i<k\\t_{i}(\vec{e})\ne\vec{e}}}\left(\mathcal{O}_{X}(q)\otimes\overline{W}^{t_{i}(\vec{e})}(\pi_{X},q)\right)\times\{\mathcal{O}_{E}(i(p-q))\}, ( O X ( q ) ⊗ W e ( π X , q ) ) × Pic 0 ( E ) ∪ 0 ≤ i < k t i ( e ) = e ⋃ ( O X ( q ) ⊗ W t i ( e ) ( π X , q ) ) × { O E ( i ( p − q ))} , endowed with the reduced structure.
As a consequence of Proposition 3.8, we obtain the equality
⋂ i = 0 k − 1 { L ∈ Pic ( C ) ∣ π ∗ ( L − ( i ⋅ p ) ) ≅ O ( σ i e ⃗ ) } = W ‾ e ⃗ ( π , p ) . \bigcap_{i=0}^{k-1}\left\{L\in\operatorname{Pic}(\mathcal{C})\mid\pi_{*}\left(L-(i\cdot p)\right)\cong\mathcal{O}(\sigma^{i}\vec{e})\right\}=\overline{\mathcal{W}}^{\vec{e}}(\pi,p). i = 0 ⋂ k − 1 { L ∈ Pic ( C ) ∣ π ∗ ( L − ( i ⋅ p ) ) ≅ O ( σ i e ) } = W e ( π , p ) . This gives a complete description of the closure and we obtain the following criterion.
Corollary 3.10. Let [ π , p ] [\pi,p] [ π , p ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot e ⃗ , f ⃗ \vec{e},\vec{f} e , f k k k
a ( e ⃗ ) ≤ a ( f ⃗ ) entrywise ⟺ W f ⃗ ( π , p ) ⊆ W ‾ e ⃗ ( π , p ) . a(\vec{e})\le a(\vec{f})\ \text{entrywise}\quad\Longleftrightarrow\quad W^{\vec{f}}(\pi,p)\subseteq\overline{W}^{\vec{e}}(\pi,p). a ( e ) ≤ a ( f ) entrywise ⟺ W f ( π , p ) ⊆ W e ( π , p ) . We are now ready to show Cohen–Macaulayness by an induction on g g g
Lemma 3.11 ([15 Lemma 7.7] and [12 Proposition 4.1] ). Let A A A B B B A ∩ B A \cap B A ∩ B 1 1 1 A ∪ B A \cup B A ∪ B A ∩ B A \cap B A ∩ B
Theorem 3.12. Let [ π : C ⟶ P 1 , p ] ∈ H g , k t o t [\pi:C \longrightarrow\mathbb{P}^{1},p] \in\mathcal{H}^{\mathrm{tot}}_{g,k} [ π : C ⟶ P 1 , p ] ∈ H g , k tot e ⃗ ∈ Z ≥ 0 k \vec{e} \in\mathbb{Z}_{\geq0}^{k} e ∈ Z ≥ 0 k W ‾ e ⃗ ( π , p ) \overline{W}^{\vec{e}}(\pi,p) W e ( π , p )
Proof. We argue by induction on g g g 1 1 1
Assume the assertion holds in genus g − 1 g-1 g − 1 k : 1 k:1 k : 1
( O X ( q ) ⊗ W ‾ e ⃗ ( π X , q ) ) × Pic 0 ( E ) \left(\mathcal{O}_{X}(q) \otimes\overline{W}^{\vec{e}}(\pi_{X},q)\right) \times\operatorname{Pic}^{0}(E) ( O X ( q ) ⊗ W e ( π X , q ) ) × Pic 0 ( E ) and
( O X ( q ) ⊗ W ‾ t i ( e ⃗ ) ( π X , q ) ) × { O E ( i ( p − q ) ) } , t i ( e ⃗ ) ≠ e ⃗ . \left(\mathcal{O}_{X}(q) \otimes\overline{W}^{t_{i}(\vec{e})}(\pi_{X},q)\right) \times\left\{\mathcal{O}_{E}\bigl(i(p-q)\bigr)\right\}, \qquad t_{i}(\vec{e}) \ne\vec{e}. ( O X ( q ) ⊗ W t i ( e ) ( π X , q ) ) × { O E ( i ( p − q ) ) } , t i ( e ) = e . All these schemes are Cohen–Macaulay by the induction hypothesis. Moreover, the scheme-theoretic inclusion
W ‾ e ⃗ ( π X , q ) ⊆ W ‾ t i ( e ⃗ ) ( π X , q ) \overline{W}^{\vec{e}}(\pi_{X},q) \subseteq\overline{W}^{t_{i}(\vec{e})}(\pi_{X},q) W e ( π X , q ) ⊆ W t i ( e ) ( π X , q ) implies that the intersection of the first scheme with the piece indexed by i i i
( O X ( q ) ⊗ W ‾ e ⃗ ( π X , q ) ) × { O E ( i ( p − q ) ) } . \left(\mathcal{O}_{X}(q) \otimes\overline{W}^{\vec{e}}(\pi_{X},q)\right) \times\left\{\mathcal{O}_{E}\bigl(i(p-q)\bigr)\right\}. ( O X ( q ) ⊗ W e ( π X , q ) ) × { O E ( i ( p − q ) ) } . This intersection is Cohen–Macaulay by induction. When nonempty, it has pure dimension
g − 1 − u ( e ⃗ ) − inv ( e ⃗ ) . g-1-u(\vec{e})-\operatorname{inv}(\vec{e}). g − 1 − u ( e ) − inv ( e ) . whereas both schemes being intersected have pure dimension
g − u ( e ⃗ ) − inv ( e ⃗ ) . g-u(\vec{e})-\operatorname{inv}(\vec{e}). g − u ( e ) − inv ( e ) . Here we use
u ( t i ( e ⃗ ) ) + inv ( t i ( e ⃗ ) ) = u ( e ⃗ ) + inv ( e ⃗ ) − 1 u\bigl(t_{i}(\vec{e})\bigr)+\operatorname{inv}\bigl(t_{i}(\vec{e})\bigr)=u(\vec{e})+\operatorname{inv}(\vec{e})-1 u ( t i ( e ) ) + inv ( t i ( e ) ) = u ( e ) + inv ( e ) − 1 for t i ( e ⃗ ) ≠ e ⃗ t_{i}(\vec{e}) \ne\vec{e} t i ( e ) = e
Since p − q p-q p − q k k k i i i [15 Lemma 7.7] shows that the special fiber is Cohen–Macaulay. Since being Cohen–Macaulay is an open condition on the base B B B W ‾ e ⃗ ( π , p ) \overline{W}^{\vec{e}}(\pi,p) W e ( π , p )
Irreducibility and Monodromy of Pointed Hurwitz–Brill–Noether loci
We note that these results have already appeared in [2 ] in terms of permutations in the k k k [2 ] , we have
W ‾ e ⃗ ( π , p ) = W τ ( C , p ) , \overline{W}^{\vec{e}}(\pi,p)=W^{\tau}(C,p), W e ( π , p ) = W τ ( C , p ) , where τ \tau τ k k k
τ ( i ) = − k e i − i for 0 ≤ i ≤ k − 1. \tau(i)=-ke_{i}-i \quad\text{for } 0 \leq i \leq k-1. τ ( i ) = − k e i − i for 0 ≤ i ≤ k − 1. Now, irreducibility and monodromy are simply reformulations of [2 Theorem 1.1] .
Proposition 3.13. Let [ π : C ⟶ P 1 , p ] ∈ H g , k t o t [\pi:C \longrightarrow\mathbb{P}^{1},p] \in\mathcal{H}^{\mathrm{tot}}_{g,k} [ π : C ⟶ P 1 , p ] ∈ H g , k tot e ⃗ ∈ Z ≥ 0 k \vec{e} \in\mathbb{Z}_{\geq0}^{k} e ∈ Z ≥ 0 k g − inv ( e ⃗ ) − u ( e ⃗ ) > 0 g-\operatorname{inv}(\vec{e})-u(\vec{e})>0 g − inv ( e ) − u ( e ) > 0 W ‾ e ⃗ ( π , p ) \overline{W}^{\vec{e}}(\pi,p) W e ( π , p )
Proposition 3.14. Let e ⃗ ∈ Z ≥ 0 k \vec{e} \in\mathbb{Z}_{\geq0}^{k} e ∈ Z ≥ 0 k g − inv ( e ⃗ ) − u ( e ⃗ ) = 0 g-\operatorname{inv}(\vec{e})-u(\vec{e})=0 g − inv ( e ) − u ( e ) = 0 W g e ⃗ W^{\vec{e}}_{g} W g e
[ π : C ⟶ P 1 , p , L ] [\pi:C\longrightarrow\mathbb{P}^{1},p,L] [ π : C ⟶ P 1 , p , L ] with [ π : C ⟶ P 1 , p ] ∈ H g , k t o t [\pi:C\longrightarrow\mathbb{P}^{1},p]\in\mathcal{H}^{\mathrm{tot}}_{g,k} [ π : C ⟶ P 1 , p ] ∈ H g , k tot [ L ] [L] [ L ] W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) W g e ⃗ W^{\vec{e}}_{g} W g e H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot
The algebraic Euler characteristic of pointed Hurwitz–Brill–Noether loci
We now use the degeneration constructed in the previous sections to study the algebraic Euler characteristic of pointed Hurwitz–Brill–Noether loci. Throughout this section, for a proper scheme X X X
χ ( X ) : = χ ( X , O X ) = ∑ i ≥ 0 ( − 1 ) i h i ( X , O X ) , \chi(X):=\chi(X,\mathcal{O}_{X})=\sum_{i\geq0}(-1)^{i}h^{i}(X,\mathcal{O}_{X}), χ ( X ) := χ ( X , O X ) = i ≥ 0 ∑ ( − 1 ) i h i ( X , O X ) , its algebraic Euler characteristic. The main advantage of this invariant in our setting is that it is preserved under flat proper degenerations. More precisely, let
C ⟶ P ⟶ B \mathcal{C}\longrightarrow\mathcal{P}\longrightarrow B C ⟶ P ⟶ B be the one parameter degeneration in Setup 3.1, whose generic fiber is a smooth degree k k k C 0 = X ∪ q E C_{0}=X\cup_{q}E C 0 = X ∪ q E W ‾ e ⃗ ( π , p ) ⟶ B \overline{W}^{\vec{e}}(\pi,p)\longrightarrow B W e ( π , p ) ⟶ B [ π : C ⟶ P 1 , p ] [\pi:C\longrightarrow\mathbb{P}^{1},p] [ π : C ⟶ P 1 , p ] W 0 e ⃗ ( π , p ) W^{\vec{e}}_{0}(\pi,p) W 0 e ( π , p )
χ ( W ‾ e ⃗ ( π , p ) ) = χ ( W 0 e ⃗ ( π , p ) ) . \chi(\overline{W}^{\vec{e}}(\pi,p))=\chi(W^{\vec{e}}_{0}(\pi,p)). χ ( W e ( π , p )) = χ ( W 0 e ( π , p )) . This is particularly useful in dimensions zero and one. If W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p )
χ ( W e ⃗ ( π , p ) ) = # W e ⃗ ( π , p ) , \chi(W^{\vec{e}}(\pi,p))=\#W^{\vec{e}}(\pi,p), χ ( W e ( π , p )) = # W e ( π , p ) , so the cardinality of the pointed Hurwitz–Brill–Noether locus may be computed on the special fiber. If W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p )
χ ( W e ⃗ ( π , p ) ) = 1 − p a ( W e ⃗ ( π , p ) ) , \chi(W^{\vec{e}}(\pi,p))=1-p_{a}(W^{\vec{e}}(\pi,p)), χ ( W e ( π , p )) = 1 − p a ( W e ( π , p )) , where p a p_{a} p a [2 Theorem 1.1] ), its genus is determined by the Euler characteristic of the special fiber.
It is well-known that the algebraic Euler characteristic is well-behaved with respect to intersections and unions. If A A A B B B X X X
0 ⟶ O A ∪ B ⟶ O A ⊕ O B ⟶ O A ∩ B ⟶ 0 0\longrightarrow\mathcal{O}_{A\cup B}\longrightarrow\mathcal{O}_{A}\oplus\mathcal{O}_{B}\longrightarrow\mathcal{O}_{A\cap B}\longrightarrow0 0 ⟶ O A ∪ B ⟶ O A ⊕ O B ⟶ O A ∩ B ⟶ 0 and consequently the relation
χ ( A ∪ B ) = χ ( A ) + χ ( B ) − χ ( A ∩ B ) . \chi(A\cup B)=\chi(A)+\chi(B)-\chi(A\cap B). χ ( A ∪ B ) = χ ( A ) + χ ( B ) − χ ( A ∩ B ) . We apply this formula to the equality χ ( W ‾ e ⃗ ( π , p ) ) = χ ( W 0 e ⃗ ( π , p ) ) \chi(\overline{W}^{\vec{e}}(\pi,p))=\chi(W^{\vec{e}}_{0}(\pi,p)) χ ( W e ( π , p )) = χ ( W 0 e ( π , p )) W 0 e ⃗ ( π , p ) W^{\vec{e}}_{0}(\pi,p) W 0 e ( π , p ) Theorem 3.5 and Proposition 3.9 we obtain the following.
Theorem 4.1. Let e ⃗ \vec{e} e k k k χ n ( e ⃗ ) : = χ ( W e ⃗ ( π , p ) ) \chi_{n}(\vec{e}):=\chi(W^{\vec{e}}(\pi,p)) χ n ( e ) := χ ( W e ( π , p )) [ π , p ] [\pi,p] [ π , p ] H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot g = n + u ( e ⃗ ) + inv ( e ⃗ ) g=n+u(\vec{e})+\operatorname{inv}(\vec{e}) g = n + u ( e ) + inv ( e )
I ( e ⃗ ) : = { i ∣ 0 ≤ i ≤ k − 1 and t i ( e ⃗ ) ≠ e ⃗ } . I(\vec{e}):=\{i\mid0\leq i\leq k-1\text{ and }t_{i}(\vec{e})\neq\vec{e}\}. I ( e ) := { i ∣ 0 ≤ i ≤ k − 1 and t i ( e ) = e } . Then we have the recursion
χ n ( e ⃗ ) = ∑ i ∈ I ( e ⃗ ) χ n ( t i ( e ⃗ ) ) − ∣ I ( e ⃗ ) ∣ χ n − 1 ( e ⃗ ) . \chi_n(\vec e)=\sum_{i\in I(\vec e)}\chi_n(t_i(\vec e))-\lvert I(\vec e)\rvert\chi_{n-1}(\vec e). χ n ( e ) = i ∈ I ( e ) ∑ χ n ( t i ( e )) − ∣ I ( e )∣ χ n − 1 ( e ) . Proof. This formula is an immediate consequence of the above Euler characteristic formula for union and intersection, of Proposition 2.7 , Proposition 3.9 and of the equalities
χ ( W e ⃗ ( π X , q ) × Pic 0 ( E ) ) = χ n − 1 ( e ⃗ ) ⋅ χ ( Pic 0 ( E ) ) \chi\left(W^{\vec e}(\pi_X,q)\times\operatorname{Pic}^{0}(E)\right)=\chi_{n-1}(\vec e)\cdot\chi\left(\operatorname{Pic}^{0}(E)\right) χ ( W e ( π X , q ) × Pic 0 ( E ) ) = χ n − 1 ( e ) ⋅ χ ( Pic 0 ( E ) ) and χ ( Pic 0 ( E ) ) = 0 \chi\left(\operatorname{Pic}^{0}(E)\right)=0 χ ( Pic 0 ( E ) ) = 0
Ducklings: a simple rule for studying maximal Hurwitz–Brill–Noether loci
We consider a maximal Hurwitz–Brill–Noether locus whose associated vector has the form
e ⃗ = ( a + 1 , … , a + 1 ⏟ k 1 t i m e s , a , … , a ⏟ k 2 t i m e s , 1 , … , 1 ⏟ k 3 t i m e s , 0 , … , 0 ⏟ k 4 t i m e s ) , \vec e=(\underbrace{a+1,\ldots,a+1}_{k_1\ \mathrm{times}},\underbrace{a,\ldots,a}_{k_2\ \mathrm{times}},\underbrace{1,\ldots,1}_{k_3\ \mathrm{times}},\underbrace{0,\ldots,0}_{k_4\ \mathrm{times}}), e = ( k 1 times a + 1 , … , a + 1 , k 2 times a , … , a , k 3 times 1 , … , 1 , k 4 times 0 , … , 0 ) , where a ≥ 2 a\geq2 a ≥ 2 k 1 > 0 k_1>0 k 1 > 0 k i k_i k i k 1 + k 2 + k 3 + k 4 = k k_1+k_2+k_3+k_4=k k 1 + k 2 + k 3 + k 4 = k m = k 1 + k 2 m=k_1+k_2 m = k 1 + k 2 0 < m < k 0<m<k 0 < m < k g = u ( e ⃗ ) g=u(\vec e) g = u ( e )
χ 0 ( e ⃗ ) = ∑ t i ( e ⃗ ) ≠ e ⃗ χ 0 ( t i ( e ⃗ ) ) \chi_0(\vec e)=\sum_{t_i(\vec e)\ne\vec e}\chi_0(t_i(\vec e)) χ 0 ( e ) = t i ( e ) = e ∑ χ 0 ( t i ( e )) as an elementary counting problem.
For the vector e ⃗ \vec e e a ( e ⃗ ) \mathbf a(\vec e) a ( e ) Proposition 1.16 and put a duckling in all southernmost squares having non-zero displacement (see Remark 1.17 , i.e. the ones corresponding to the first k 1 + k 2 k_1+k_2 k 1 + k 2 leading duckling to be the one whose associated position has the smallest number. We denote by Δ \Delta Δ Definition 1.11 and we can give a precise description of the final vector (i.e. the vector f ⃗ \vec f f a ( f ⃗ ) \mathbf a(\vec f) a ( f ) f ⃗ \vec f f ∣ e ⃗ ∣ + k − k 1 − k 2 + 1 , … , ∣ e ⃗ ∣ \lvert\vec e\rvert+k-k_1-k_2+1,\ldots,\lvert\vec e\rvert ∣ e ∣ + k − k 1 − k 2 + 1 , … , ∣ e ∣
Figure 5. Example of two moves in the duckling game. Under t 0 t_0 t 0 t 1 t_1 t 1
The displacement (cf. Remark 1.17 ) of the southernmost squares in a ( f ⃗ ) \mathbf a(\vec f) a ( f ) Theorem 4.1 and Proposition 1.16 .
Lemma 4.2. Let e ⃗ \vec{e} e m m m S S S { 0 , 1 , … , k − 1 } \{0,1,\ldots,k-1\} { 0 , 1 , … , k − 1 }
S : = { { 0 , … , k 1 − 1 } ∪ { k − k 2 , … , k − 1 } , when k 2 > 0 , { k − m , … , k − 1 } , when k 2 = 0. S := \begin{cases}
\{0,\ldots,k_{1}-1\}\cup\{k-k_{2},\ldots,k-1\}, & \text{when } k_{2}>0,\\
\{k-m,\ldots,k-1\}, & \text{when } k_{2}=0.
\end{cases} S := { { 0 , … , k 1 − 1 } ∪ { k − k 2 , … , k − 1 } , { k − m , … , k − 1 } , when k 2 > 0 , when k 2 = 0. Moreover, we denote
Δ : = { ( a − 1 ) ( k − m ) − k 3 , if k 2 > 0 , a ( k − m ) − k 3 , if k 2 = 0. \Delta:= \begin{cases}
(a-1)(k-m)-k_{3}, & \text{if } k_{2}>0,\\
a(k-m)-k_{3}, & \text{if } k_{2}=0.
\end{cases} Δ := { ( a − 1 ) ( k − m ) − k 3 , a ( k − m ) − k 3 , if k 2 > 0 , if k 2 = 0. We place ducklings on the line Z ≥ 0 \mathbb{Z}_{\geq0} Z ≥ 0 S S S m m m
Δ + k − m , … , Δ + k − 1 , \Delta+k-m,\ldots,\Delta+k-1, Δ + k − m , … , Δ + k − 1 , allowing only moves according to the following rules:
At each move, exactly one duckling advances by one position (moves from spot i i i i + 1 i+1 i + 1
Two ducklings never occupy the same position;
The leading duckling (i.e. positioned in the highest number) never passes Δ + k − 1 \Delta+k-1 Δ + k − 1
The distance between the leading and last duckling is always smaller than k k k
Let e ( S , Δ ) e(S,\Delta) e ( S , Δ )
e ( S , Δ ) = χ 0 ( e ⃗ ) . e(S,\Delta)=\chi_{0}(\vec{e}). e ( S , Δ ) = χ 0 ( e ) . Proof. Let f ⃗ \vec{f} f
a ( f ⃗ ) = ( 0 , 1 , … , k − 1 + ∑ i = 0 k − 1 e i ) . \mathbf{a}(\vec{f})=(0,1,\ldots,k-1+\sum_{i=0}^{k-1}e_i). a ( f ) = ( 0 , 1 , … , k − 1 + i = 0 ∑ k − 1 e i ) . Using the formula
χ 0 ( e ⃗ ) = ∑ t i ( e ⃗ ) ≠ e ⃗ χ 0 ( t i ( e ⃗ ) ) , \chi_{0}(\vec{e})=\sum_{t_i(\vec{e})\ne\vec{e}}\chi_{0}(t_i(\vec{e})), χ 0 ( e ) = t i ( e ) = e ∑ χ 0 ( t i ( e )) , together with χ 0 ( f ⃗ ) = 1 \chi_{0}(\vec{f})=1 χ 0 ( f ) = 1 χ 0 ( e ⃗ ) \chi_{0}(\vec{e}) χ 0 ( e ) t i t_i t i f ⃗ \vec{f} f t i t_i t i
We use the displacement description of Remark 1.17 . We arrange the numbers in a ( e ⃗ ) \mathbf{a}(\vec{e}) a ( e ) Proposition 1.16 and look at the southernmost squares having non-zero displacement (see Remark 1.17 , i.e. the ones corresponding to the first m m m e ⃗ \vec{e} e t i t_i t i
In order for a transformation t i t_i t i u , v u,v u , v u − 1 , v + 1 u-1,v+1 u − 1 , v + 1 u − v ≥ k + 1 u-v\geq k+1 u − v ≥ k + 1 e ⃗ \vec{e} e k − 1 k-1 k − 1
Every transformation t i t_i t i 0 0 0 1 1 1
The distance between first and last displaced southernmost squares can never be bigger than k − 1 k-1 k − 1
The order of the southernmost squares stays the same at each step and no two can be on the same position.
Finally, the leading southernmost square cannot advance more than its displacement. In particular, the ducklings in the lemma and the displaced southernmost squares of e ⃗ \vec{e} e e ( S , Δ ) = χ 0 ( e ⃗ ) e(S,\Delta)=\chi_{0}(\vec{e}) e ( S , Δ ) = χ 0 ( e ) □ \square □
Figure 6. Duckling game corresponding to Figure 4
Our next goal is to consider the case of gonality k = 4 k = 4 k = 4
Lemma 4.3. Let k = 4 k = 4 k = 4 Lemma 4.2 we have e ( { 2 , 3 } , n ) = 2 n − 1 e(\{2,3\},n)=2^{n-1} e ({ 2 , 3 } , n ) = 2 n − 1
Proof. We will compute this recursively. Since the two ducklings cannot sit on the same number, the first move is forced: the leading duckling has to move. This implies that e ( { 2 , 3 } , n ) e(\{2,3\},n) e ({ 2 , 3 } , n ) n + 2 , n + 3 n+2,n+3 n + 2 , n + 3 n + 1 , n + 2 n+1,n+2 n + 1 , n + 2
e ( { 2 , 3 } , n ) = e ( { 1 , 3 } , n − 1 ) . e(\{2,3\},n)=e(\{1,3\},n-1). e ({ 2 , 3 } , n ) = e ({ 1 , 3 } , n − 1 ) . When the ducklings are in positions 1 and 3, both of them can move, and we obtain, after translating backwards, the recursion formula
e ( { 1 , 3 } , n ) = e ( { 2 , 3 } , n ) + e ( { 0 , 3 } , n − 1 ) . e(\{1,3\},n)=e(\{2,3\},n)+e(\{0,3\},n-1). e ({ 1 , 3 } , n ) = e ({ 2 , 3 } , n ) + e ({ 0 , 3 } , n − 1 ) . But if the ducklings are in position 0 and 3, the move is forced: the leading duck cannot advance, since the distance between the ducks is always smaller than k = 4 k=4 k = 4
e ( { 0 , 3 } , n − 1 ) = e ( { 1 , 3 } , n − 1 ) . e(\{0,3\},n-1)=e(\{1,3\},n-1). e ({ 0 , 3 } , n − 1 ) = e ({ 1 , 3 } , n − 1 ) . In particular, we obtain e ( { 1 , 3 } , n ) = 2 e ( { 1 , 3 } , n − 1 ) e(\{1,3\},n)=2e(\{1,3\},n-1) e ({ 1 , 3 } , n ) = 2 e ({ 1 , 3 } , n − 1 ) e ( { 1 , 3 } , 0 ) = 1 e(\{1,3\},0)=1 e ({ 1 , 3 } , 0 ) = 1 e ( { 1 , 3 } , n ) = 2 n e(\{1,3\},n)=2^n e ({ 1 , 3 } , n ) = 2 n
This has implications for the computations of cardinalities of Hurwitz–Brill–Noether loci.
Corollary 4.4. Let [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^{1},p] [ π : C → P 1 , p ] H g , 4 t o t \mathcal{H}^{\mathrm{tot}}_{g,4} H g , 4 tot g g g e ⃗ \vec{e} e W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p )
Table 4.4.
In fact, keeping track of the position of the displaced southernmost square is sufficient also when we want to compute the algebraic Euler characteristic for higher dimensional Hurwitz–Brill–Noether loci.
Let e ⃗ \vec{e} e Δ + k − 1 − s \Delta+k-1-s Δ + k − 1 − s s ∈ S s\in S s ∈ S e n ( S , Δ ) e_n(S,\Delta) e n ( S , Δ ) χ n ( e ⃗ ) \chi_n(\vec{e}) χ n ( e ) e n ( S , Δ ) e_n(S,\Delta) e n ( S , Δ ) e ⃗ \vec{e} e σ \sigma σ Definition 1.8 . We have χ n ( e ⃗ ) = χ n ( σ e ⃗ ) \chi_n(\vec{e})=\chi_n(\sigma\vec{e}) χ n ( e ) = χ n ( σ e ) σ \sigma σ O C ( p ) \mathcal{O}_C(p) O C ( p )
We will outline again how to compute algebraic Euler characteristics in gonality 4.
Lemma 4.5. Let k = 4 k=4 k = 4 n ≥ 1 n\geq1 n ≥ 1 e 1 ( { 2 , 3 } , n ) = − ( 3 n − 1 ) 2 n − 1 e_1(\{2,3\},n)=-(3n-1)2^{n-1} e 1 ({ 2 , 3 } , n ) = − ( 3 n − 1 ) 2 n − 1
Proof. The formula in Theorem 4.1 implies
e 1 ( { 2 , 3 } , n ) = e 1 ( { 1 , 3 } , n − 1 ) − e 0 ( { 2 , 3 } , n ) = e 1 ( { 1 , 3 } , n − 1 ) − 2 n − 1 . e_1(\{2,3\},n)=e_1(\{1,3\},n-1)-e_0(\{2,3\},n)=e_1(\{1,3\},n-1)-2^{n-1}. e 1 ({ 2 , 3 } , n ) = e 1 ({ 1 , 3 } , n − 1 ) − e 0 ({ 2 , 3 } , n ) = e 1 ({ 1 , 3 } , n − 1 ) − 2 n − 1 . Furthermore, applying the same formula for e 1 ( { 1 , 3 } , n ) e_1(\{1,3\},n) e 1 ({ 1 , 3 } , n )
e 1 ( { 1 , 3 } , n ) = e 1 ( { 2 , 3 } , n ) + e 1 ( { 0 , 3 } , n − 1 ) − 2 e 0 ( { 1 , 3 } , n ) = e 1 ( { 2 , 3 } , n ) + e 1 ( { 0 , 3 } , n − 1 ) − 2 n + 1 . e_1(\{1,3\},n)=e_1(\{2,3\},n)+e_1(\{0,3\},n-1)-2e_0(\{1,3\},n)=e_1(\{2,3\},n)+e_1(\{0,3\},n-1)-2^{n+1}. e 1 ({ 1 , 3 } , n ) = e 1 ({ 2 , 3 } , n ) + e 1 ({ 0 , 3 } , n − 1 ) − 2 e 0 ({ 1 , 3 } , n ) = e 1 ({ 2 , 3 } , n ) + e 1 ({ 0 , 3 } , n − 1 ) − 2 n + 1 . Similarly we obtain
e 1 ( { 0 , 3 } , n ) = e 1 ( { 1 , 3 } , n ) − e 0 ( { 0 , 3 } , n ) = e 1 ( { 1 , 3 } , n ) − 2 n . e_1(\{0,3\},n)=e_1(\{1,3\},n)-e_0(\{0,3\},n)=e_1(\{1,3\},n)-2^n. e 1 ({ 0 , 3 } , n ) = e 1 ({ 1 , 3 } , n ) − e 0 ({ 0 , 3 } , n ) = e 1 ({ 1 , 3 } , n ) − 2 n . In particular, out of the last two formulas, we obtain
e 1 ( { 1 , 3 } , n ) = e 1 ( { 2 , 3 } , n ) + e 1 ( { 1 , 3 } , n − 1 ) − 2 n − 1 − 2 n + 1 . e_1(\{1,3\},n)=e_1(\{2,3\},n)+e_1(\{1,3\},n-1)-2^{n-1}-2^{n+1}. e 1 ({ 1 , 3 } , n ) = e 1 ({ 2 , 3 } , n ) + e 1 ({ 1 , 3 } , n − 1 ) − 2 n − 1 − 2 n + 1 . Therefore
e 1 ( { 1 , 3 } , n ) = 2 e 1 ( { 1 , 3 } , n − 1 ) − 2 n − 1 − 2 n − 1 − 2 n + 1 = 2 e 1 ( { 1 , 3 } , n − 1 ) − 3 ⋅ 2 n . e_1(\{1,3\},n)=2e_1(\{1,3\},n-1)-2^{n-1}-2^{n-1}-2^{n+1}=2e_1(\{1,3\},n-1)-3\cdot2^n. e 1 ({ 1 , 3 } , n ) = 2 e 1 ({ 1 , 3 } , n − 1 ) − 2 n − 1 − 2 n − 1 − 2 n + 1 = 2 e 1 ({ 1 , 3 } , n − 1 ) − 3 ⋅ 2 n . Using the initial condition
e 1 ( { 1 , 3 } , 0 ) = e 1 ( { 2 , 3 } , 0 ) − e 0 ( { 1 , 3 } , 0 ) = 0 − 1 = − 1 , e_1(\{1,3\},0)=e_1(\{2,3\},0)-e_0(\{1,3\},0)=0-1=-1, e 1 ({ 1 , 3 } , 0 ) = e 1 ({ 2 , 3 } , 0 ) − e 0 ({ 1 , 3 } , 0 ) = 0 − 1 = − 1 , we obtain
e 1 ( { 1 , 3 } , n ) = − ( 3 n + 1 ) 2 n . e_1(\{1,3\},n)=-(3n+1)2^n. e 1 ({ 1 , 3 } , n ) = − ( 3 n + 1 ) 2 n . As a consequence of this result, we obtain the genus of maximal pointed Hurwitz–Brill–Noether loci, when their dimension is 1 and the gonality is 4.
Corollary 4.6. Let [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^1,p] [ π : C → P 1 , p ] H g , 4 t o t \mathcal{H}^{\mathrm{tot}}_{g,4} H g , 4 tot g g g
Table 4.6.
Low codimension existence
For a nonnegative k k k e ⃗ \vec{e} e
W g e ⃗ : = { [ π : C ⟶ P 1 , p , L ] | [ π , p ] ∈ H g , k t o t , L ∈ W e ⃗ ( π , p ) } . W^{\vec{e}}_g:=\left\{[\pi:C\longrightarrow\mathbb{P}^1,p,L]\ \middle|\ [\pi,p]\in\mathcal{H}^{\mathrm{tot}}_{g,k},\ L\in W^{\vec{e}}(\pi,p)\right\}. W g e := { [ π : C ⟶ P 1 , p , L ] [ π , p ] ∈ H g , k tot , L ∈ W e ( π , p ) } . This is a locally closed subscheme of the relative Picard space of degree g + k + ∣ e ⃗ ∣ − 1 g+k+|\vec{e}|-1 g + k + ∣ e ∣ − 1 H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot
W g e ⃗ ⟶ H g , k t o t W^{\vec{e}}_g\longrightarrow\mathcal{H}^{\mathrm{tot}}_{g,k} W g e ⟶ H g , k tot forgets L L L W e ⃗ ( π , p ) W^{\vec{e}}(\pi,p) W e ( π , p ) [ π , p ] [\pi,p] [ π , p ] H ( e ⃗ ) \mathcal{H}(\vec{e}) H ( e )
We start with the case of a degree k k k π : C → P 1 \pi:C\to\mathbb{P}^1 π : C → P 1 g g g p p p k ≥ g + 1 k\geq g+1 k ≥ g + 1 H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot M g , 1 \mathcal{M}_{g,1} M g , 1 H ( e ⃗ ) \mathcal{H}(\vec{e}) H ( e ) H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot u ( e ⃗ ) + inv ( e ⃗ ) − g u(\vec{e})+\operatorname{inv}(\vec{e})-g u ( e ) + inv ( e ) − g M g \mathcal{M}_g M g
Proposition 5.1. Consider integers g g g r r r d d d k k k
k ≥ g + 1 , r ≥ 1 , 2 r ≤ d ≤ g + r − 2 , 3 − g ≤ ρ ( g , r , d ) ≤ 0 , k \ge g+1,\qquad r \ge1,\qquad2r \le d \le g+r-2,\qquad3-g \le\rho(g,r,d) \le0, k ≥ g + 1 , r ≥ 1 , 2 r ≤ d ≤ g + r − 2 , 3 − g ≤ ρ ( g , r , d ) ≤ 0 , and set
e ⃗ = ( 2 , … , 2 ⏟ r + 1 times , 1 , … , 1 ⏟ k − g − 1 + d − 2 r times , 0 , … , 0 ⏟ g − d + r times ) . \vec{e}=\left(\underbrace{2,\ldots,2}_{r+1\text{ times}},\underbrace{1,\ldots,1}_{k-g-1+d-2r\text{ times}},\underbrace{0,\ldots,0}_{g-d+r\text{ times}}\right). e = r + 1 times 2 , … , 2 , k − g − 1 + d − 2 r times 1 , … , 1 , g − d + r times 0 , … , 0 . Then W g e ⃗ \mathcal{W}^{\vec{e}}_g W g e 2 g + k − 3 + g − u ( e ⃗ ) − inv ( e ⃗ ) 2g+k-3+g-u(\vec{e})-\operatorname{inv}(\vec{e}) 2 g + k − 3 + g − u ( e ) − inv ( e ) H g , k tot \mathcal{H}^{\operatorname{tot}}_{g,k} H g , k tot M g , d r \mathcal{M}^r_{g,d} M g , d r u ( e ⃗ ) + inv ( e ⃗ ) − g u(\vec{e})+\operatorname{inv}(\vec{e})-g u ( e ) + inv ( e ) − g H g , k tot \mathcal{H}^{\operatorname{tot}}_{g,k} H g , k tot
For r = 1 r=1 r = 1 2 − g ≤ ρ ( g , 1 , d ) ≤ 0 2-g\le\rho(g,1,d)\le0 2 − g ≤ ρ ( g , 1 , d ) ≤ 0
Proof. The entries of e ⃗ \vec{e} e ∣ e ⃗ ∣ = k − g + d + 1 \lvert\vec{e}\rvert=k-g+d+1 ∣ e ∣ = k − g + d + 1 W g e ⃗ \mathcal{W}^{\vec{e}}_g W g e
g + k + ∣ e ⃗ ∣ − 1 = d + 2 k , g+k+\lvert\vec{e}\rvert-1=d+2k, g + k + ∣ e ∣ − 1 = d + 2 k , and their prescribed vanishing sequence is
a ( e ⃗ ) = ( 0 , 1 , … , 2 k − g + d − r − 1 , 2 k , 2 k + 1 , … , 2 k + r ) . a(\vec{e})=(0,1,\ldots,2k-g+d-r-1,2k,2k+1,\ldots,2k+r). a ( e ) = ( 0 , 1 , … , 2 k − g + d − r − 1 , 2 k , 2 k + 1 , … , 2 k + r ) . Moreover, since e ⃗ \vec{e} e
inv ( e ⃗ ) = 0 and u ( e ⃗ ) = ( r + 1 ) ( g − d + r ) , \operatorname{inv}(\vec{e})=0\quad\text{and}\quad u(\vec{e})=(r+1)(g-d+r), inv ( e ) = 0 and u ( e ) = ( r + 1 ) ( g − d + r ) , and hence g − ρ ( g , r , d ) = g − u ( e ⃗ ) − inv ( e ⃗ ) g-\rho(g,r,d)=g-u(\vec{e})-\operatorname{inv}(\vec{e}) g − ρ ( g , r , d ) = g − u ( e ) − inv ( e )
In order to prove our result, we will use the Brill–Noether counterpart proved in [18 Theorem A] and pull it back to H g , k tot \mathcal{H}^{\operatorname{tot}}_{g,k} H g , k tot
W e ⃗ ( π : C → P 1 , p ) → ⊗ O C ( − 2 k p ) W d r , ∘ ( C ) \mathcal{W}^{\vec{e}}(\pi:C\to\mathbb{P}^{1},p)\xrightarrow{\ \otimes\mathcal{O}_{C}(-2kp)\ }\mathcal{W}^{r,\circ}_{d}(C) W e ( π : C → P 1 , p ) ⊗ O C ( − 2 k p ) W d r , ∘ ( C ) where W d r , ∘ ( C ) \mathcal{W}^{r,\circ}_{d}(C) W d r , ∘ ( C ) M M M h 0 ( C , M ) = r + 1 h^{0}(C,M)=r+1 h 0 ( C , M ) = r + 1 M M M ω C ⊗ M − 1 \omega_C\otimes M^{-1} ω C ⊗ M − 1 p p p p p p
( 0 , 1 , … , r ) and ( 0 , 1 , … , g − d + r − 1 ) , (0,1,\ldots,r)\quad\text{and}\quad(0,1,\ldots,g-d+r-1), ( 0 , 1 , … , r ) and ( 0 , 1 , … , g − d + r − 1 ) , respectively. All these conditions ensure that the vanishing orders of M ( 2 k p ) M(2kp) M ( 2 k p ) p p p a ( e ⃗ ) a(\vec{e}) a ( e )
We consider the universal Brill–Noether space
W g , d r , ∘ = { [ C , L ] ∣ C ∈ M g , L ∈ Pic d ( C ) , and h 0 ( C , L ) = r + 1 } . \mathcal{W}^{r,\circ}_{g,d}=\{[C,L]\mid C\in\mathcal{M}_g,\ L\in\operatorname{Pic}^{d}(C),\ \text{and }h^{0}(C,L)=r+1\}. W g , d r , ∘ = {[ C , L ] ∣ C ∈ M g , L ∈ Pic d ( C ) , and h 0 ( C , L ) = r + 1 } . Thus twisting by O C ( 2 k p ) \mathcal{O}_{C}(2kp) O C ( 2 k p ) W g e ⃗ \mathcal{W}^{\vec{e}}_g W g e
H g , k tot × M g W g , d r , ∘ \mathcal{H}^{\operatorname{tot}}_{g,k}\mathop{\times}_{\mathcal{M}_g}\mathcal{W}^{r,\circ}_{g,d} H g , k tot × M g W g , d r , ∘ where p p p W g , d r , ∘ ( C ) \mathcal{W}^{r,\circ}_{g,d}(C) W g , d r , ∘ ( C )
We now construct the required component. By [18 Theorem A] , W g , d r , ∘ \mathcal{W}^{r,\circ}_{g,d} W g , d r , ∘ Z Z Z 3 g − 3 + ρ ( g , r , d ) 3g-3+\rho(g,r,d) 3 g − 3 + ρ ( g , r , d ) M g \mathcal{M}_g M g
H g , k tot , ∘ × M g Z → M g , 1 × M g Z → Z ↓ ↓ ↓ H g , k tot , ∘ → M g , 1 ∘ → M g \begin{CD}
\mathcal{H}^{\operatorname{tot},\circ}_{g,k}\mathop{\times}_{\mathcal{M}_g} Z @>>> \mathcal{M}_{g,1}\mathop{\times}_{\mathcal{M}_g} Z @>>> Z\\
@VVV @VVV @VVV\\
\mathcal{H}^{\operatorname{tot},\circ}_{g,k} @>>> \mathcal{M}^{\circ}_{g,1} @>>> \mathcal{M}_g
\end{CD} H g , k tot , ∘ × M g Z ↓ ⏐ H g , k tot , ∘ M g , 1 × M g Z ↓ ⏐ M g , 1 ∘ Z ↓ ⏐ M g Here we denoted by M g , 1 ∘ \mathcal{M}^{\circ}_{g,1} M g , 1 ∘ M g , 1 \mathcal{M}_{g,1} M g , 1 [ C , p ] [C,p] [ C , p ] p p p ∣ M ∣ |M| ∣ M ∣ ∣ ω C ⊗ M − 1 ∣ |\omega_C\otimes M^{-1}| ∣ ω C ⊗ M − 1 ∣ M M M Z → M g Z\to\mathcal{M}_g Z → M g [ C ] [C] [ C ] H g , k tot , ∘ \mathcal{H}^{\operatorname{tot},\circ}_{g,k} H g , k tot , ∘ p ∈ C p \in C p ∈ C ω C \omega_C ω C k ≥ g + 1 k \ge g+1 k ≥ g + 1
h 0 ( C , O C ( k p ) ) = k − g + 1 , h 0 ( C , O C ( ( k − 1 ) p ) ) = k − g . h^0(C,\mathcal{O}_C(kp))=k-g+1,\qquad h^0(C,\mathcal{O}_C((k-1)p))=k-g. h 0 ( C , O C ( k p )) = k − g + 1 , h 0 ( C , O C (( k − 1 ) p )) = k − g . The degree k k k p p p
P ( H 0 ( C , O C ( k p ) ) / C ) ∖ P ( H 0 ( C , O C ( ( k − 1 ) p ) ) / C ) . \mathbb{P}(H^0(C,\mathcal{O}_C(kp))/\mathbb{C})\setminus\mathbb{P}(H^0(C,\mathcal{O}_C((k-1)p))/\mathbb{C}). P ( H 0 ( C , O C ( k p )) / C ) ∖ P ( H 0 ( C , O C (( k − 1 ) p )) / C ) . Indeed, such a pencil is spanned by the constant function and a function whose unique pole has order exactly k k k p p p k − g − 1 k-g-1 k − g − 1
H g , k t o t , ∘ ⟶ M g , 1 ∘ \mathcal{H}^{\mathrm{tot},\circ}_{g,k}\longrightarrow\mathcal{M}^{\circ}_{g,1} H g , k tot , ∘ ⟶ M g , 1 ∘ has fibers of dimension k − g − 1 k-g-1 k − g − 1
In particular, we obtain
dim H g , k t o t , ∘ × M g Z = 3 g − 2 + k − g − 1 + ρ ( g , r , d ) = 2 g + k − 3 + g − u ( e ⃗ ) − inv ( e ⃗ ) . \dim\mathcal{H}^{\mathrm{tot},\circ}_{g,k}\mathbin{\times_{\mathcal{M}_g}} Z
=3g-2+k-g-1+\rho(g,r,d)
=2g+k-3+g-u(\vec e)-\operatorname{inv}(\vec e). dim H g , k tot , ∘ × M g Z = 3 g − 2 + k − g − 1 + ρ ( g , r , d ) = 2 g + k − 3 + g − u ( e ) − inv ( e ) . This is a component of W g e ⃗ \mathcal{W}^{\vec e}_g W g e H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot [ π : C → P 1 , p ] [\pi:C\to\mathbb{P}^1,p] [ π : C → P 1 , p ] [ π : C → P 1 , p , L ] ∈ W g e ⃗ [\pi:C\to\mathbb{P}^1,p,L]\in\mathcal{W}^{\vec e}_g [ π : C → P 1 , p , L ] ∈ W g e L ( − 2 k ⋅ p ) ∈ W d r ( C ) L(-2k\cdot p)\in W^r_d(C) L ( − 2 k ⋅ p ) ∈ W d r ( C ) Z Z Z
Thus we obtained a component of W g e ⃗ \mathcal{W}^{\vec e}_g W g e
For the case r = 1 r=1 r = 1 2 − g = ρ ( g , 1 , d ) 2-g=\rho(g,1,d) 2 − g = ρ ( g , 1 , d ) d = 2 d=2 d = 2 g − 2 g-2 g − 2 M g \mathcal{M}_g M g
A similar result can be obtained for pointed Hurwitz–Brill–Noether subspaces of H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot [18 Theorem A] .
Proposition 5.2. Consider integers g g g r r r d d d k k k
k ≥ g + 1 , r ≥ 1 , 2 r ≤ d ≤ g + r − 2 , r + 1 − g ≤ ρ ( g , r , d ) ≤ 0 , k\ge g+1,\qquad r\ge1,\qquad2r\le d\le g+r-2,\qquad r+1-g\le\rho(g,r,d)\le0, k ≥ g + 1 , r ≥ 1 , 2 r ≤ d ≤ g + r − 2 , r + 1 − g ≤ ρ ( g , r , d ) ≤ 0 , and set
e ⃗ = ( 2 , … , 2 ⏟ r + 1 times , 1 , … , 1 ⏟ k − g − 1 + d − 2 r times , 0 , … , 0 ⏟ g − d + r times ) . \vec e=\left(
\underbrace{2,\ldots,2}_{r+1\ \text{times}},
\underbrace{1,\ldots,1}_{k-g-1+d-2r\ \text{times}},
\underbrace{0,\ldots,0}_{g-d+r\ \text{times}}
\right). e = r + 1 times 2 , … , 2 , k − g − 1 + d − 2 r times 1 , … , 1 , g − d + r times 0 , … , 0 . Then H g ( e ⃗ ) \mathcal{H}_g(\vec e) H g ( e ) 2 g + k − 3 + g − u ( e ⃗ ) − inv ( e ⃗ ) 2g+k-3+g-u(\vec e)-\operatorname{inv}(\vec e) 2 g + k − 3 + g − u ( e ) − inv ( e ) [ π , p ] [\pi,p] [ π , p ] W e ⃗ ( π , p ) \mathcal{W}^{\vec e}(\pi,p) W e ( π , p ) 0 0 0
Proof. There exists Z Z Z M g , d r \mathcal{M}^r_{g,d} M g , d r [ C ] ∈ Z [C]\in Z [ C ] ∈ Z
dim W d r ( C ) = 0 and h 0 ( C , L ) = r + 1 for all L ∈ W d r ( C ) . \dim W^r_d(C)=0\quad\text{and}\quad h^0(C,L)=r+1\quad\text{for all }L\in W^r_d(C). dim W d r ( C ) = 0 and h 0 ( C , L ) = r + 1 for all L ∈ W d r ( C ) . A generic element in the preimage of Z Z Z H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot H g ( e ⃗ ) \mathcal{H}_g(\vec e) H g ( e ) M g \mathcal{M}_g M g M g , d r \mathcal{M}^r_{g,d} M g , d r Z Z Z Z Z Z
Our main focus will be on the (pullback of the) hyperelliptic locus. We will use an inductive argument to reduce the existence of a pointed Hurwitz–Brill–Noether component of the expected dimension to the known fact that the hyperelliptic locus has expected dimension.
Proposition 5.3. Let e ⃗ = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k \vec e=(e_0,\ldots,e_{k-1})\in\mathbb{Z}_{\ge0}^k e = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k
1 ≤ c : = u ( e ⃗ ) + inv ( e ⃗ ) − g ≤ k − 3 , ∣ e ⃗ ∣ ≡ 1 − c ( m o d k ) . 1\le c:=u(\vec e)+\operatorname{inv}(\vec e)-g\le k-3,\qquad|\vec e|\equiv1-c\pmod{k}. 1 ≤ c := u ( e ) + inv ( e ) − g ≤ k − 3 , ∣ e ∣ ≡ 1 − c ( mod k ) . Write
b 0 ( e ⃗ ) < b 1 ( e ⃗ ) < ⋯ < b k − 1 ( e ⃗ ) b_0(\vec e)<b_1(\vec e)<\cdots<b_{k-1}(\vec e) b 0 ( e ) < b 1 ( e ) < ⋯ < b k − 1 ( e ) for the increasing rearrangement of the integers k e j + j ke_j + j k e j + j 0 ≤ j ≤ k − 1 0 \le j \le k - 1 0 ≤ j ≤ k − 1 b k − 2 − b c > k b_{k-2} - b_c > k b k − 2 − b c > k W g e ⃗ \mathcal{W}^{\vec e}_g W g e 2 g + k − 3 − c 2g + k - 3 - c 2 g + k − 3 − c H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot
Proof. Set s = ∣ e ⃗ ∣ + c − 1 − k k s = \frac{|\vec e|+c-1-k}{k} s = k ∣ e ∣ + c − 1 − k
f ⃗ = ( s + 2 , s + 2 , s + 1 , … , s + 1 ⏟ k − c − 3 times , s , … , s ⏟ c + 1 times ) . \vec f = (s+2,s+2,\underbrace{s+1,\ldots,s+1}_{k-c-3\text{ times}},\underbrace{s,\ldots,s}_{c+1\text{ times}}). f = ( s + 2 , s + 2 , k − c − 3 times s + 1 , … , s + 1 , c + 1 times s , … , s ) . We will show that the locus W c + 2 f ⃗ \mathcal{W}^{\vec f}_{c+2} W c + 2 f H c + 2 , k t o t \mathcal{H}^{\mathrm{tot}}_{c+2,k} H c + 2 , k tot W g − 1 t j ( e ⃗ ) \mathcal{W}^{t_j(\vec e)}_{g-1} W g − 1 t j ( e ) W g e ⃗ \mathcal{W}^{\vec e}_g W g e t j t_j t j Definition 1.11 ) such that t j ( e ⃗ ) ≠ e ⃗ t_j(\vec e) \ne\vec e t j ( e ) = e e ⃗ \vec e e f ⃗ \vec f f g − c − 2 g-c-2 g − c − 2
We begin with the base case, which is the pullback of the hyperelliptic locus.
Claim 4. The locus W c + 2 f ⃗ \mathcal{W}^{\vec f}_{c+2} W c + 2 f k + c + 1 k+c+1 k + c + 1 H c + 2 , k t o t \mathcal{H}^{\mathrm{tot}}_{c+2,k} H c + 2 , k tot
Proof of Claim 4. In the notations at the start of the proof, we consider the pointed Hurwitz–Brill–Noether space W c + 2 f ⃗ \mathcal{W}^{\vec f}_{c+2} W c + 2 f c + 2 c+2 c + 2
∣ f ⃗ ∣ = ∣ e ⃗ ∣ , u ( f ⃗ ) = 2 ( c + 1 ) , inv ( f ⃗ ) = 0. |\vec f| = |\vec e|,\qquad u(\vec f) = 2(c+1),\qquad\operatorname{inv}(\vec f) = 0. ∣ f ∣ = ∣ e ∣ , u ( f ) = 2 ( c + 1 ) , inv ( f ) = 0. Thus f ⃗ \vec f f c + 2 c+2 c + 2 s s s u + inv u+\operatorname{inv} u + inv P 1 \mathbb{P}^1 P 1 Proposition 5.1 , the locus W c + 2 f ⃗ \mathcal{W}^{\vec f}_{c+2} W c + 2 f k + c + 1 k+c+1 k + c + 1 H c + 2 , k t o t \mathcal{H}^{\mathrm{tot}}_{c+2,k} H c + 2 , k tot
We now explain the inductive strategy.
Claim 5. Suppose that t j ( e ⃗ ) ≠ e ⃗ t_j(\vec e) \ne\vec e t j ( e ) = e g − 1 g-1 g − 1 W g − 1 t j ( e ⃗ ) \mathcal{W}^{t_j(\vec e)}_{g-1} W g − 1 t j ( e ) Z Z Z 2 ( g − 1 ) + k − 3 + ( g − u ( e ⃗ ) − inv ( e ⃗ ) ) 2(g-1)+k-3+(g-u(\vec e)-\operatorname{inv}(\vec e)) 2 ( g − 1 ) + k − 3 + ( g − u ( e ) − inv ( e )) H g − 1 , k t o t \mathcal{H}^{\mathrm{tot}}_{g-1,k} H g − 1 , k tot W g e ⃗ \mathcal{W}^{\vec e}_g W g e 3 g − 3 − u ( e ⃗ ) − inv ( e ⃗ ) 3g-3-u(\vec e)-\operatorname{inv}(\vec e) 3 g − 3 − u ( e ) − inv ( e ) H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot
Proof of Claim 5. By Proposition 1.16 , we have
u ( t j ( e ⃗ ) ) + inv ( t j ( e ⃗ ) ) = u ( e ⃗ ) + inv ( e ⃗ ) − 1. u(t_j(\vec e))+\operatorname{inv}(t_j(\vec e))=u(\vec e)+\operatorname{inv}(\vec e)-1. u ( t j ( e )) + inv ( t j ( e )) = u ( e ) + inv ( e ) − 1. Therefore
u ( t j ( e ⃗ ) ) + inv ( t j ( e ⃗ ) ) − ( g − 1 ) = c , u(t_j(\vec e))+\operatorname{inv}(t_j(\vec e))-(g-1)=c, u ( t j ( e )) + inv ( t j ( e )) − ( g − 1 ) = c , so the value of the expected codimension is unchanged when we decrease the genus by one and apply a nontrivial elementary shift.
We show that the conclusion of the theorem then holds for e ⃗ \vec e e g g g [ π X : X ⟶ P 1 , q , L ] ∈ Z [\pi_X:X\longrightarrow\mathbb{P}^1,q,L]\in Z [ π X : X ⟶ P 1 , q , L ] ∈ Z E E E q q q p p p p − q p-q p − q k k k
π 0 : X ∪ q E ⟶ P 1 ∪ P 1 , \pi_0:X\cup_q E\longrightarrow\mathbb{P}^1\cup\mathbb{P}^1, π 0 : X ∪ q E ⟶ P 1 ∪ P 1 , totally ramified at point p p p
C ⟶ P ⟶ B , C\longrightarrow P\longrightarrow B, C ⟶ P ⟶ B , a versal deformation around a sufficiently small neighborhood B B B [ π 0 , p ] [\pi_0,p] [ π 0 , p ] H ‾ g , k t o t \overline{\mathcal{H}}^{\mathrm{tot}}_{g,k} H g , k tot B B B 2 g + k − 3 2g+k-3 2 g + k − 3 Δ ⊂ B \Delta\subset B Δ ⊂ B g g g g − 1 g-1 g − 1
Theorem 3.4 can be extended to arbitrary families of admissible k : 1 k:1 k : 1 Δ \Delta Δ ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e ) e ⃗ \vec{e} e Proposition 2.2 , first proof)
( O X ( q ) ⊗ W g − 1 e ⃗ ) × Pic 0 ( E → Δ ) , (\mathcal{O}_{X}(q) \otimes W^{\vec{e}}_{g-1}) \times\operatorname{Pic}^{0}(\mathcal{E} \to\Delta), ( O X ( q ) ⊗ W g − 1 e ) × Pic 0 ( E → Δ ) , and
( O X ( q ) ⊗ W g − 1 t i ( e ⃗ ) ) × { O E ( i ⋅ p − i ⋅ q ) } for 0 ≤ i ≤ k − 1 satisfying t i ( e ⃗ ) ≠ e ⃗ . (\mathcal{O}_{X}(q) \otimes W^{t_i(\vec{e})}_{g-1}) \times\{\mathcal{O}_{E}(i \cdot p-i \cdot q)\}\text{ for }0 \leq i \leq k-1\text{ satisfying }t_i(\vec{e}) \ne\vec{e}. ( O X ( q ) ⊗ W g − 1 t i ( e ) ) × { O E ( i ⋅ p − i ⋅ q )} for 0 ≤ i ≤ k − 1 satisfying t i ( e ) = e . By Lemma 3.1 and Lemma 3.2 , all elements in these loci contain a unique ord ( e ⃗ ) \operatorname{ord}(\vec{e}) ord ( e )
( O X ( q ) ⊗ Z ) × { O E ( j ⋅ p − j ⋅ q ) } (\mathcal{O}_{X}(q) \otimes Z) \times\{\mathcal{O}_{E}(j \cdot p-j \cdot q)\} ( O X ( q ) ⊗ Z ) × { O E ( j ⋅ p − j ⋅ q )} has dimension
2 ( g − 1 ) + k − 3 + g − u ( e ⃗ ) − inv ( e ⃗ ) + 1 = 3 g − 4 + k − u ( e ⃗ ) − inv ( e ⃗ ) , 2(g-1)+k-3+g-u(\vec{e})-\operatorname{inv}(\vec{e})+1=3g-4+k-u(\vec{e})-\operatorname{inv}(\vec{e}), 2 ( g − 1 ) + k − 3 + g − u ( e ) − inv ( e ) + 1 = 3 g − 4 + k − u ( e ) − inv ( e ) , a similar Regeneration Theorem to Theorem 3.4 implies it is contained in the closure of W g e ⃗ \mathcal{W}^{\vec{e}}_{g} W g e Y Y Y W g e ⃗ \mathcal{W}^{\vec{e}}_{g} W g e Y → H g , k t o t Y \to\mathcal{H}^{\mathrm{tot}}_{g,k} Y → H g , k tot [ π 0 , p ] [\pi_0,p] [ π 0 , p ] 0 0 0 Z Z Z 0 0 0
( O X ( q ) ⊗ Z ) × { O E ( j ⋅ p − j ⋅ q ) } (\mathcal{O}_{X}(q) \otimes Z) \times\{\mathcal{O}_{E}(j \cdot p-j \cdot q)\} ( O X ( q ) ⊗ Z ) × { O E ( j ⋅ p − j ⋅ q )} is cut out from Y Y Y t 1 = 0 t_1=0 t 1 = 0
dim Y = 1 + dim ( ( O X ( q ) ⊗ Z ) × { O E ( j ⋅ p − j ⋅ q ) } ) = dim Z + 2 , \dim Y=1+\dim\bigl((\mathcal{O}_{X}(q) \otimes Z)\times\{\mathcal{O}_{E}(j\cdot p-j\cdot q)\}\bigr)=\dim Z+2, dim Y = 1 + dim ( ( O X ( q ) ⊗ Z ) × { O E ( j ⋅ p − j ⋅ q )} ) = dim Z + 2 , as required. Consequently, W g e ⃗ \mathcal{W}^{\vec{e}}_{g} W g e 3 g − 3 + k − u ( e ⃗ ) − inv ( e ⃗ ) 3g-3+k-u(\vec{e})-\operatorname{inv}(\vec{e}) 3 g − 3 + k − u ( e ) − inv ( e ) H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot
To apply the inductive argument, it remains only to prove that e ⃗ \vec{e} e f ⃗ \vec{f} f Definition 1.11 .
Given a vector d ⃗ = ( d 0 , … , d k − 1 ) ∈ Z ≥ 0 k \vec{d}=(d_0,\ldots,d_{k-1})\in\mathbb{Z}_{\geq0}^{k} d = ( d 0 , … , d k − 1 ) ∈ Z ≥ 0 k b 0 ( d ⃗ ) < ⋯ < b k − 1 ( d ⃗ ) b_0(\vec{d})<\cdots<b_{k-1}(\vec{d}) b 0 ( d ) < ⋯ < b k − 1 ( d ) k d j + j kd_j+j k d j + j 0 ≤ j ≤ k − 1 0\leq j\leq k-1 0 ≤ j ≤ k − 1 d ⃗ \vec{d} d
Claim 6. There is a finite sequence of elementary shifts t j 1 ( 1 ) , … , t j g − c − 2 ( g − c − 2 ) t_{j_1}^{(1)},\ldots,t_{j_{g-c-2}}^{(g-c-2)} t j 1 ( 1 ) , … , t j g − c − 2 ( g − c − 2 )
f ⃗ = t j g − c − 2 ( g − c − 2 ) ∘ ⋯ ∘ t j 1 ( 1 ) ( e ⃗ ) , \vec{f}=t_{j_{g-c-2}}^{(g-c-2)}\circ\cdots\circ t_{j_1}^{(1)}(\vec{e}), f = t j g − c − 2 ( g − c − 2 ) ∘ ⋯ ∘ t j 1 ( 1 ) ( e ) , and at each intermediate step (for every 0 ≤ ℓ ≤ g − c − 2 0\leq\ell\leq g-c-2 0 ≤ ℓ ≤ g − c − 2 d ⃗ = t j ℓ ( ℓ ) ∘ ⋯ ∘ t j 1 ( 1 ) ( e ⃗ ) \vec{d}=t_{j_\ell}^{(\ell)}\circ\cdots\circ t_{j_1}^{(1)}(\vec{e}) d = t j ℓ ( ℓ ) ∘ ⋯ ∘ t j 1 ( 1 ) ( e ) t j 0 ( 0 ) = id t_{j_0}^{(0)}=\operatorname{id} t j 0 ( 0 ) = id t j ℓ + 1 ( ℓ + 1 ) ( d ⃗ ) ≠ d ⃗ t_{j_{\ell+1}}^{(\ell+1)}(\vec{d})\ne\vec{d} t j ℓ + 1 ( ℓ + 1 ) ( d ) = d
(i) b k − 2 ( d ⃗ ) − b c ( d ⃗ ) > k b_{k-2}(\vec{d})-b_c(\vec{d})>k b k − 2 ( d ) − b c ( d ) > k
(ii) 1 ≤ c = u ( d ⃗ ) + inv ( d ⃗ ) − ( g − ℓ ) ≤ k − 3 1\leq c=u(\vec{d})+\operatorname{inv}(\vec{d})-(g-\ell)\leq k-3 1 ≤ c = u ( d ) + inv ( d ) − ( g − ℓ ) ≤ k − 3
(iii) ∣ d ⃗ ∣ ≡ 1 − c ( m o d k ) |\vec{d}|\equiv1-c\pmod{k} ∣ d ∣ ≡ 1 − c ( mod k )
are satisfied.
Proof of Claim 6. Starting from d ⃗ \vec{d} d t j t_j t j u ( d ⃗ ) + inv ( d ⃗ ) u(\vec{d})+\operatorname{inv}(\vec{d}) u ( d ) + inv ( d ) 1 1 1
1 ≤ c : = u ( d ⃗ ) + inv ( d ⃗ ) − ( g − ℓ ) ≤ k − 3 , ∣ d ⃗ ∣ ≡ 1 − c ( m o d k ) 1\leq c:=u(\vec{d})+\operatorname{inv}(\vec{d})-(g-\ell)\leq k-3,\qquad|\vec{d}|\equiv1-c\pmod{k} 1 ≤ c := u ( d ) + inv ( d ) − ( g − ℓ ) ≤ k − 3 , ∣ d ∣ ≡ 1 − c ( mod k ) are automatically satisfied, as the sum of the entries is unchanged after applying t j t_j t j g g g
u ( d ⃗ ) + inv ( d ⃗ ) − ( g − ℓ ) = u ( t j ( d ⃗ ) ) + inv ( t j ( d ⃗ ) ) − ( g − ℓ − 1 ) . u(\vec{d})+\operatorname{inv}(\vec{d})-(g-\ell)=u(t_j(\vec{d}))+\operatorname{inv}(t_j(\vec{d}))-(g-\ell-1). u ( d ) + inv ( d ) − ( g − ℓ ) = u ( t j ( d )) + inv ( t j ( d )) − ( g − ℓ − 1 ) . We turn our focus to (i), namely that we can choose some t j t_j t j
b k − 2 ( d ⃗ ) − b c ( d ⃗ ) > k . (1) b_{k-2}(\vec{d})-b_c(\vec{d})>k. \tag*{(1)} b k − 2 ( d ) − b c ( d ) > k . (1) For a vector ( d 0 , … , d k − 1 ) (d_0,\ldots,d_{k-1}) ( d 0 , … , d k − 1 ) j j j k d j + j kd_j+j k d j + j 1 ≤ j ≤ k − 1 1\leq j\leq k-1 1 ≤ j ≤ k − 1 d j > d j − 1 d_j>d_{j-1} d j > d j − 1 t j t_j t j
k d j − 1 + j − 1 , k d j + j kd_{j-1}+j-1,\qquad kd_j+j k d j − 1 + j − 1 , k d j + j with
k d j − 1 + j , k d j + j − 1. kd_{j-1}+j,\qquad kd_j+j-1. k d j − 1 + j , k d j + j − 1. Thus the larger of these two numbers moves one square to the left and the smaller moves one square to the right. Similarly, when t 0 t_0 t 0 d 0 > d k − 1 + 1 d_0>d_{k-1}+1 d 0 > d k − 1 + 1
k d k − 1 + k − 1 , k d 0 kd_{k-1}+k-1,\qquad kd_0 k d k − 1 + k − 1 , k d 0 are replaced by
k d k − 1 + k , k d 0 − 1. kd_{k-1}+k,\qquad kd_0-1. k d k − 1 + k , k d 0 − 1. Also in this case, the larger of these two numbers moves one square to the left and the smaller moves one square to the right, cyclically.
Note that at each step, at most one nontrivial transformation can fail to preserve (1) . Indeed, the difference in (1) changes by at most two after each application of a t j t_j t j k k k k k k (1) can fail only when
b k − 2 ( d ⃗ ) − b c ( d ⃗ ) = k + 1. b_{k-2}(\vec{d})-b_c(\vec{d})=k+1. b k − 2 ( d ) − b c ( d ) = k + 1. and the transform replaces these two endpoints by b k − 2 ( d ⃗ ) − 1 b_{k-2}(\vec{d})-1 b k − 2 ( d ) − 1 b c ( d ⃗ ) + 1 b_c(\vec{d})+1 b c ( d ) + 1
We now show that the required t j ℓ t_j^\ell t j ℓ
Starting from e ⃗ \vec{e} e (1) , as long as one exists. Every move decreases u + inv u+\operatorname{inv} u + inv t j t_j t j k k k (1) . Thus at most one transformation can violate (1) , and therefore the stopping vector has exactly one nontrivial transformation. It is the transformation changing b k − 2 b_{k-2} b k − 2 b c b_c b c b k − 2 − 1 b_{k-2}-1 b k − 2 − 1 b c + 1 b_c+1 b c + 1
It remains to show that the number of t j t_j t j g − c − 2 g-c-2 g − c − 2 f ⃗ \vec{f} f
Now let d ⃗ \vec{d} d (1) is t j t_j t j 0 ≤ j < k 0\leq j<k 0 ≤ j < k σ j \sigma^j σ j − j -j − j t 0 t_0 t 0
Since all transformations t i t_i t i 1 ≤ i ≤ k − 1 1\leq i\leq k-1 1 ≤ i ≤ k − 1 n ⃗ = σ j d ⃗ \vec{n}=\sigma^j\vec{d} n = σ j d n 0 ≥ n 1 ≥ ⋯ ≥ n k − 1 n_0\geq n_1\geq\cdots\geq n_{k-1} n 0 ≥ n 1 ≥ ⋯ ≥ n k − 1 t 0 t_0 t 0 k − 2 k-2 k − 2 c c c k + 1 k+1 k + 1 b c ( n ⃗ ) = k n k − 1 + k − 1 b_c(\vec{n})=kn_{k-1}+k-1 b c ( n ) = k n k − 1 + k − 1 b k − 2 ( n ⃗ ) = k n 0 b_{k-2}(\vec{n})=kn_0 b k − 2 ( n ) = k n 0 k n 0 − ( k n k − 1 + k − 1 ) = k + 1 kn_0-(kn_{k-1}+k-1)=k+1 k n 0 − ( k n k − 1 + k − 1 ) = k + 1
n 0 − n k − 1 = 2 , n_0-n_{k-1}=2, n 0 − n k − 1 = 2 , whereby n 0 − n i ≤ 2 n_0-n_i\leq2 n 0 − n i ≤ 2 0 ≤ i ≤ k − 1 0\leq i\leq k-1 0 ≤ i ≤ k − 1 b k − 2 ( n ⃗ ) = k n 0 b_{k-2}(\vec{n})=kn_0 b k − 2 ( n ) = k n 0 b k − 1 ( n ⃗ ) ≥ b k − 2 ( n ⃗ ) b_{k-1}(\vec{n})\geq b_{k-2}(\vec{n}) b k − 1 ( n ) ≥ b k − 2 ( n ) 0 ≤ i ≤ k − 1 0\leq i\leq k-1 0 ≤ i ≤ k − 1 b k − 1 ( n ⃗ ) = k n i + i > k n 0 = b k − 2 ( n ⃗ ) b_{k-1}(\vec{n})=kn_i+i>kn_0=b_{k-2}(\vec{n}) b k − 1 ( n ) = k n i + i > k n 0 = b k − 2 ( n ) i ≥ 2 i\geq2 i ≥ 2 k ( n 0 − n i ) < i ≤ k − 1 k(n_0-n_i)<i\leq k-1 k ( n 0 − n i ) < i ≤ k − 1 n 0 = n i n_0=n_i n 0 = n i n ⃗ \vec{n} n i i i i = 1 i=1 i = 1 n 0 = n 1 > n i ≥ n 0 − 2 n_0=n_1>n_i\geq n_0-2 n 0 = n 1 > n i ≥ n 0 − 2 2 ≤ i ≤ k − 1 2\leq i\leq k-1 2 ≤ i ≤ k − 1
Since b c ( n ⃗ ) = k n k − 1 + k − 1 b_c(\vec{n})=kn_{k-1}+k-1 b c ( n ) = k n k − 1 + k − 1 c c c b 0 ( n ⃗ ) = k n i 1 + i 1 < ⋯ < b c − 1 ( n ⃗ ) = k n i c + i c b_0(\vec{n})=kn_{i_1}+i_1<\cdots<b_{c-1}(\vec{n})=kn_{i_c}+i_c b 0 ( n ) = k n i 1 + i 1 < ⋯ < b c − 1 ( n ) = k n i c + i c b c ( n ⃗ ) = k n k − 1 + k − 1 b_c(\vec{n})=kn_{k-1}+k-1 b c ( n ) = k n k − 1 + k − 1 n i j ≠ n k − 1 n_{i_j}\ne n_{k-1} n i j = n k − 1 n i j = n k − 1 + 1 n_{i_j}=n_{k-1}+1 n i j = n k − 1 + 1
k n i j + i j = k ( n k − 1 + 1 ) + i j = k n k − 1 + k + i j < k n k − 1 + k − 1 , kn_{i_j}+i_j=k(n_{k-1}+1)+i_j=kn_{k-1}+k+i_j<kn_{k-1}+k-1, k n i j + i j = k ( n k − 1 + 1 ) + i j = k n k − 1 + k + i j < k n k − 1 + k − 1 , which gives i j < 0 i_j<0 i j < 0 n i j = n k − 1 n_{i_j}=n_{k-1} n i j = n k − 1
n ⃗ = ( n 0 , n 0 , n 0 − 1 , … , n 0 − 1 ⏟ k − c − 3 t i m e s , n 0 − 2 , … , n 0 − 2 ⏟ c + 1 t i m e s ) . \vec{n}=(n_0,n_0,\underbrace{n_0-1,\ldots,n_0-1}_{k-c-3\ \mathrm{times}},\underbrace{n_0-2,\ldots,n_0-2}_{c+1\ \mathrm{times}}). n = ( n 0 , n 0 , k − c − 3 times n 0 − 1 , … , n 0 − 1 , c + 1 times n 0 − 2 , … , n 0 − 2 ) . Directly computing, we see that ∣ n ⃗ ∣ ≡ 1 − c ( m o d k ) |\vec{n}|\equiv1-c\pmod{k} ∣ n ∣ ≡ 1 − c ( mod k ) n ⃗ = σ j d ⃗ \vec{n}=\sigma^j\vec{d} n = σ j d
∣ n ⃗ ∣ = ∣ σ j d ⃗ ∣ = ∣ d ⃗ ∣ − j = ∣ e ⃗ ∣ − j ≡ 1 − c − j ( m o d k ) . |\vec{n}|=|\sigma^j\vec{d}|=|\vec{d}|-j=|\vec{e}|-j\equiv1-c-j\pmod{k}. ∣ n ∣ = ∣ σ j d ∣ = ∣ d ∣ − j = ∣ e ∣ − j ≡ 1 − c − j ( mod k ) . Consequently
1 − c − j ≡ 1 − c ( m o d k ) . 1-c-j\equiv1-c\pmod{k}. 1 − c − j ≡ 1 − c ( mod k ) . Since 0 ≤ j < k 0\leq j<k 0 ≤ j < k j = 0 j=0 j = 0 n ⃗ \vec{n} n
k ( n 0 − 2 ) + 4 + k − c − 3 = ∣ n ⃗ ∣ = ∣ e ⃗ ∣ = k s + k + 1 − c , k(n_0-2)+4+k-c-3=|\vec{n}|=|\vec{e}|=ks+k+1-c, k ( n 0 − 2 ) + 4 + k − c − 3 = ∣ n ∣ = ∣ e ∣ = k s + k + 1 − c , we have s = n 0 − 2 s=n_0-2 s = n 0 − 2 n ⃗ = f ⃗ \vec{n}=\vec{f} n = f
u ( e ⃗ ) + inv ( e ⃗ ) − u ( f ⃗ ) − inv ( f ⃗ ) = g + c − 2 ( c + 1 ) = g − c − 2. u(\vec{e})+\operatorname{inv}(\vec{e})-u(\vec{f})-\operatorname{inv}(\vec{f})=g+c-2(c+1)=g-c-2. u ( e ) + inv ( e ) − u ( f ) − inv ( f ) = g + c − 2 ( c + 1 ) = g − c − 2. Finally, we are in a position to use the induction to conclude. Starting with the component constructed in Claim 4 and using the induction of Claim 5 for each of the g − c − 2 g-c-2 g − c − 2 Claim 6 , we obtain a component of W g e ⃗ \mathcal{W}^{\vec{e}}_g W g e
( k + c + 1 ) + 2 ( g − c − 2 ) = 2 g + k − 3 − c , (k+c+1)+2(g-c-2)=2g+k-3-c, ( k + c + 1 ) + 2 ( g − c − 2 ) = 2 g + k − 3 − c , generically finite over its image in H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot
Notice that condition ∣ e ⃗ ∣ ≡ 1 − c ( m o d k ) |\vec{e}|\equiv1-c\pmod{k} ∣ e ∣ ≡ 1 − c ( mod k ) Definition 1.8
σ − 1 e ⃗ = ( e k − 1 + 1 , e 0 , e 1 , … , e k − 2 ) , \sigma^{-1}\vec{e}=(e_{k-1}+1,e_0,e_1,\ldots,e_{k-2}), σ − 1 e = ( e k − 1 + 1 , e 0 , e 1 , … , e k − 2 ) , we have W σ − 1 e ⃗ ( π , p ) = O C ( p ) ⊗ W e ⃗ ( π , p ) \mathcal{W}^{\sigma^{-1}\vec{e}}(\pi,p)=\mathcal{O}_C(p)\otimes\mathcal{W}^{\vec{e}}(\pi,p) W σ − 1 e ( π , p ) = O C ( p ) ⊗ W e ( π , p ) [ π , p ] ∈ H g , k t o t [\pi,p]\in\mathcal{H}^{\mathrm{tot}}_{g,k} [ π , p ] ∈ H g , k tot O C ( p ) \mathcal{O}_C(p) O C ( p ) Proposition 5.3 we can take c c c u ( e ⃗ ) + inv ( e ⃗ ) − g u(\vec{e})+\operatorname{inv}(\vec{e})-g u ( e ) + inv ( e ) − g σ − 1 \sigma^{-1} σ − 1 Proposition 5.3 . We thus immediately obtain the following.
Theorem 5.4. Let g ≥ 1 g\geq1 g ≥ 1 e ⃗ = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k \vec{e}=(e_0,\ldots,e_{k-1})\in\mathbb{Z}_{\geq0}^k e = ( e 0 , … , e k − 1 ) ∈ Z ≥ 0 k
1 ≤ c : = u ( e ⃗ ) + inv ( e ⃗ ) − g ≤ k − 3. 1\leq c:=u(\vec{e})+\operatorname{inv}(\vec{e})-g\leq k-3. 1 ≤ c := u ( e ) + inv ( e ) − g ≤ k − 3. Write b 0 ( e ⃗ ) < b 1 ( e ⃗ ) < ⋯ < b k − 1 ( e ⃗ ) b_0(\vec{e})<b_1(\vec{e})<\cdots<b_{k-1}(\vec{e}) b 0 ( e ) < b 1 ( e ) < ⋯ < b k − 1 ( e ) k e j + j ke_j+j k e j + j 0 ≤ j ≤ k − 1 0\leq j\leq k-1 0 ≤ j ≤ k − 1 b k − 2 ( e ⃗ ) − b c ( e ⃗ ) > k b_{k-2}(\vec{e})-b_c(\vec{e})>k b k − 2 ( e ) − b c ( e ) > k W g e ⃗ \mathcal{W}^{\vec{e}}_g W g e 2 g + k − 3 + ( g − u ( e ⃗ ) − inv ( e ⃗ ) ) 2g+k-3+(g-u(\vec{e})-\operatorname{inv}(\vec{e})) 2 g + k − 3 + ( g − u ( e ) − inv ( e )) H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot
The expected codimension k − 3 k-3 k − 3 k − 2 k-2 k − 2 k − 3 k-3 k − 3 g = k − 1 g=k-1 g = k − 1 H g , k t o t → M g , 1 \mathcal{H}^{\mathrm{tot}}_{g,k}\to\mathcal{M}_{g,1} H g , k tot → M g , 1 M g , 1 \mathcal{M}_{g,1} M g , 1 H g ( 2 g − 2 ) \mathcal{H}_g(2g-2) H g ( 2 g − 2 ) [ C , p ] [C,p] [ C , p ] ω C ≅ O C ( ( 2 g − 2 ) p ) \omega_C\cong\mathcal{O}_C((2g-2)p) ω C ≅ O C (( 2 g − 2 ) p ) H g ( 2 g − 2 ) \mathcal{H}_g(2g-2) H g ( 2 g − 2 ) H g h y p ( 2 g − 2 ) \mathcal{H}^{\mathrm{hyp}}_g(2g-2) H g hyp ( 2 g − 2 ) H g o d d ( 2 g − 2 ) \mathcal{H}^{\mathrm{odd}}_g(2g-2) H g odd ( 2 g − 2 ) H g e v e n ( 2 g − 2 ) \mathcal{H}^{\mathrm{even}}_g(2g-2) H g even ( 2 g − 2 ) [13 ] . For a generic element [ C , p ] [C,p] [ C , p ] ω C \omega_C ω C p p p [3 ] , and are given as
( 0 , 2 , 4 , … , 2 g − 2 ) (0,2,4,\ldots,2g-2) ( 0 , 2 , 4 , … , 2 g − 2 ) H g h y p ( 2 g − 2 ) \mathcal{H}^{\mathrm{hyp}}_g(2g-2) H g hyp ( 2 g − 2 )
( 0 , 1 , … , g − 2 , 2 g − 2 ) (0,1,\ldots,g-2,2g-2) ( 0 , 1 , … , g − 2 , 2 g − 2 ) H g o d d ( 2 g − 2 ) \mathcal{H}^{\mathrm{odd}}_g(2g-2) H g odd ( 2 g − 2 )
( 0 , 1 , … , g − 3 , g − 1 , 2 g − 2 ) (0,1,\ldots,g-3,g-1,2g-2) ( 0 , 1 , … , g − 3 , g − 1 , 2 g − 2 ) H g e v e n ( 2 g − 2 ) \mathcal{H}^{\mathrm{even}}_g(2g-2) H g even ( 2 g − 2 )
But knowing all the vanishing orders uniquely determines the vector of the corresponding Hurwitz–Brill–Noether locus, giving us the following results.
Lemma 5.5. Let [ C , p ] [C,p] [ C , p ] H g ( 2 g − 2 ) \mathcal{H}_g(2g-2) H g ( 2 g − 2 ) π : C → P 1 \pi:C\to\mathbb{P}^1 π : C → P 1 ( g + 1 ) (g+1) ( g + 1 ) p p p
(1) If [ C , p ] ∈ H g o d d ( 2 g − 2 ) [C,p]\in\mathcal{H}^{\mathrm{odd}}_g(2g-2) [ C , p ] ∈ H g odd ( 2 g − 2 ) ω C ( 2 ( g + 1 ) p ) ∈ W → o d d ( π , p ) \omega_C(2(g+1)p)\in\overrightarrow{W}^{\mathrm{odd}}(\pi,p) ω C ( 2 ( g + 1 ) p ) ∈ W odd ( π , p )
e ⃗ o d d = ( 0 , … , 0 ⏟ ( g − 3 ) times , 1 , 0 , − 1 , − 2 ) + ( 2 , 2 , … , 2 ) \vec{e}_{\mathrm{odd}}=(\underbrace{0,\ldots,0}_{(g-3)\ \text{times}},1,0,-1,-2)+(2,2,\ldots,2) e odd = ( ( g − 3 ) times 0 , … , 0 , 1 , 0 , − 1 , − 2 ) + ( 2 , 2 , … , 2 ) (2) If [ C , p ] ∈ H g e v e n ( 2 g − 2 ) [C,p]\in\mathcal{H}^{\mathrm{even}}_g(2g-2) [ C , p ] ∈ H g even ( 2 g − 2 ) ω C ( 2 ( g + 1 ) p ) ∈ W → e v e n ( π , p ) \omega_C(2(g+1)p)\in\overrightarrow{W}^{\mathrm{even}}(\pi,p) ω C ( 2 ( g + 1 ) p ) ∈ W even ( π , p )
e ⃗ e v e n = ( 0 , … , 0 ⏟ ( g − 3 ) times , 1 , − 1 , 0 , − 2 ) + ( 2 , 2 , … , 2 ) \vec{e}_{\mathrm{even}}=(\underbrace{0,\ldots,0}_{(g-3)\ \text{times}},1,-1,0,-2)+(2,2,\ldots,2) e even = ( ( g − 3 ) times 0 , … , 0 , 1 , − 1 , 0 , − 2 ) + ( 2 , 2 , … , 2 ) Corollary 5.6. Let g ≥ 4 g\ge4 g ≥ 4 k = g + 1 k=g+1 k = g + 1 W → g e v e n \overrightarrow{W}^{\mathrm{even}}_g W g even 2 g 2g 2 g
Proof. A simple calculation gives
u ( e ⃗ e v e n ) = g + 1 , and inv ( e ⃗ e v e n ) = g − 2. u(\vec{e}_{\mathrm{even}})=g+1,\quad\text{and}\quad\operatorname{inv}(\vec{e}_{\mathrm{even}})=g-2. u ( e even ) = g + 1 , and inv ( e even ) = g − 2. In particular, the expected codimension of W → g e v e n \overrightarrow{W}^{\mathrm{even}}_g W g even H g , k t o t \mathcal{H}^{\mathrm{tot}}_{g,k} H g , k tot g − 1 g-1 g − 1 H g , k t o t → M g , 1 \mathcal{H}^{\mathrm{tot}}_{g,k}\to\mathcal{M}_{g,1} H g , k tot → M g , 1 [ C , p ] [C,p] [ C , p ]
P ( H 0 ( C , O C ( ( g + 1 ) p ) ) / C ) ∖ P ( H 0 ( C , O C ( g p ) ) / C ) \mathbb{P}\left(H^0(C,\mathcal{O}_C((g+1)p))/\mathbb{C}\right)\setminus\mathbb{P}\left(H^0(C,\mathcal{O}_C(gp))/\mathbb{C}\right) P ( H 0 ( C , O C (( g + 1 ) p )) / C ) ∖ P ( H 0 ( C , O C ( g p )) / C ) and this is one dimensional for [ C , p ] [C,p] [ C , p ] H g e v e n ( 2 g − 2 ) \mathcal{H}^{\mathrm{even}}_g(2g-2) H g even ( 2 g − 2 ) H g , k t o t → M g , 1 \mathcal{H}^{\mathrm{tot}}_{g,k}\to\mathcal{M}_{g,1} H g , k tot → M g , 1 H g e v e n ( 2 g − 2 ) \mathcal{H}^{\mathrm{even}}_g(2g-2) H g even ( 2 g − 2 )
dim W → g e v e n = dim H g e v e n ( 2 g − 2 ) + 1 = 2 g . \dim\overrightarrow{W}^{\mathrm{even}}_g=\dim\mathcal{H}^{\mathrm{even}}_g(2g-2)+1=2g. dim W g even = dim H g even ( 2 g − 2 ) + 1 = 2 g . But, when k = g + 1 k=g+1 k = g + 1 dim H g , k t o t = 3 g − 2 \dim\mathcal{H}^{\mathrm{tot}}_{g,k}=3g-2 dim H g , k tot = 3 g − 2
References
Bud, Institut Mathématique de Toulouse; Université de Toulouse; CNRS, 118 route de Narbonne, 31062 Toulouse, France Email address: andreibud95@protonmail.com
Haburcak, Department of Mathematics, The Ohio State University, 100 Math Tower, Columbus, OH 43210 Email address: haburcak.1@osu.edu
