Introduction
Conic bundles occur naturally in the birational classification of complex threefolds.1 [16 ] for an account of this theory. Inspired by the recently introduced theory of Hodge atoms of Katzarkov–Kontsevich–Pantev–Yu [13 ] , we study this problem through the algebraic structure of big quantum cohomology.
A standard conic bundle is a surjective projective flat morphism π : X → S \pi: X \to S π : X → S ρ ( X / S ) = 1 \rho(X/S) = 1 ρ ( X / S ) = 1 Δ ⊂ S \Delta\subset S Δ ⊂ S S S S
Δ S : = 2 K S + Δ . (1) \Delta_S := 2K_S + \Delta. \tag*{(1)} Δ S := 2 K S + Δ. (1) Shokurov conjectured that the effectivity of Δ S \Delta_S Δ S [17 Conjecture 10.3] ; see also [7 Conjecture 2.2] , and an equivalent conjecture in [12 ] . Here we prove this implication:
Theorem A. Let π : X → S \pi: X \to S π : X → S Δ \Delta Δ
∣ Δ S ∣ ≠ ∅ , \lvert\Delta_S \rvert\ne\varnothing, ∣ Δ S ∣ = ∅ , then X X X
Theorem A is first proven in the case where Δ S \Delta_S Δ S
The quantum calculation
1.1. The quantum calculation. In [13 ] , Katzarkov–Kontsevich–Pantev–Yu combine Hodge theory with Gromov–Witten theory to construct Hodge atoms. Their construction relates the spectral decomposition of quantum multiplication by the Euler vector field, along the locus of Hodge classes, to the Hodge-theoretic structures carried by its summands. The behavior of these objects under blowups provides obstructions to rationality.
For our purposes, we use the formal algebraic approach of [5 ] . We study quantum multiplication κ ′ = E u t ′ ⋆ t ′ ( − ) \kappa' = E u_{t'} \star_{t'} (-) κ ′ = E u t ′ ⋆ t ′ ( − ) K X \mathbb{K}_X K X § 2 . The relevant obstruction is that, for a smooth projective rational threefold, every eigenvalue of this operator has algebraic multiplicity at most two [5 Theorem 5.6] . Here we produce an eigenvalue of larger multiplicity by constructing a local Artinian factor:
Theorem B. Let π : X → S \pi: X \to S π : X → S Δ S \Delta_S Δ S ρ S : = rank NS ( S ) \rho_S := \operatorname{rank} \operatorname{NS}(S) ρ S := rank NS ( S )
Q H e v e n ( X ) ⊗ N o v r e d K X QH^{\mathrm{even}}(X) \otimes_{\mathrm{Nov}^{\mathrm{red}}} K_X Q H even ( X ) ⊗ Nov red K X has a local Artinian factor H − H_- H −
dim K X H − = ρ S + 2 , dim K X rad ( H − ) = ρ S + 1 , \dim_{K_X} H_- = \rho_S + 2,\qquad\dim_{K_X} \operatorname{rad}(H_-) = \rho_S + 1, dim K X H − = ρ S + 2 , dim K X rad ( H − ) = ρ S + 1 , dim K X ( rad ( H − ) ) 2 = 1 , rad ( H − ) 3 = 0. \dim_{K_X}\left(\operatorname{rad}(H_-)\right)^2 = 1,\qquad\operatorname{rad}(H_-)^3 = 0. dim K X ( rad ( H − ) ) 2 = 1 , rad ( H − ) 3 = 0. The restriction of κ t ′ \kappa_{t'} κ t ′ H − H_- H − ρ S + 2 \rho_S + 2 ρ S + 2 X X X
The proof of Theorem B has two steps. First, we compute the vertical quantum product , obtained by retaining only curve classes contracted by π \pi π [11 §5] . After extending coefficients and inverting q ℓ q^\ell q ℓ ℓ \ell ℓ H e v e n ( S , Q ) H^{\mathrm{even}}(S,\mathbb{Q}) H even ( S , Q )
Second, we include the contributions of nonvertical curves and lift this decomposition to the full quantum product. The main point is to prove that one of the two lifted summands remains local over the algebraically closed fraction field. Nefness of Δ S \Delta_S Δ S κ t ′ \kappa_{t'} κ t ′ ρ S + 2 \rho_S + 2 ρ S + 2
Birational consequences
Theorem A, together with the known results, gives the rationality criterion formulated in [16 Conjecture 1.2] . Here J ( X ) J(X) J ( X )
Corollary C. Let π : X → S \pi: X \to S π : X → S Δ \Delta Δ X X X ∣ 2 K S + Δ ∣ = ∅ \lvert2K_S + \Delta\rvert= \varnothing ∣ 2 K S + Δ ∣ = ∅ J ( X ) J(X) J ( X )
As a last contribution, we apply the calculation to nodal quartic double solids , that is, double covers of P 3 \mathbb{P}^3 P 3 Q \mathbb{Q} Q
Corollary D. Let Y Y Y N N N
Y is rational ⟺ Y is not Q -factorial and N ≠ 6. Y\text{ is rational}\quad\Longleftrightarrow\quad Y\text{ is not }\mathbb{Q}\text{-factorial and }N \ne6. Y is rational ⟺ Y is not Q -factorial and N = 6. Cheltsov–Przyjalkowski–Shramov established the connection between nodal quartic double solids and Shokurov’s conjecture in [7 Corollary 4.28] . For a Q \mathbb{Q} Q N ≥ 1 N \ge1 N ≥ 1 Δ ∼ − 2 K S \Delta\sim-2K_S Δ ∼ − 2 K S ρ S = N \rho_S = N ρ S = N N + 2 N + 2 N + 2 Q \mathbb{Q} Q [7 Theorems 1.2 and 1.5] . Moreover, Clemens’ criterion expresses Q \mathbb{Q} Q P 3 \mathbb{P}^3 P 3 [7 Theorem 1.6] . Building on that, in Section 5, we give a rationality test in terms of a matrix-rank condition.
These conclusions fit into a long history of irrationality results for quartic double solids. Artin–Mumford constructed a ten-nodal example whose smooth resolution has nonzero torsion in H 3 ( − , Z ) H^3(-,\mathbb{Z}) H 3 ( − , Z ) [2 ] . Beauville’s Prym-variety approach gives the one-node irrationality result and explains the limitation of the intermediate-Jacobian obstruction for examples with at least seven nodes [3 Example 4.10.4] . For smooth quartic double solids, the intermediate-Jacobian approach was further developed by Tikhomirov [18 ] and Voisin [19 ] . More recently, using decomposition of the diagonal, Voisin proved stable irrationality for very general double solids with at most seven nodes in the families specified in [20 Theorem 1.1 and Remark 1.2] . Achter–Casalaina-Martin–Vial extended this to cases with at most nine nodes [1 Theorem 3 and Section 17] .
This paper is the starting point for a new direction, patching together the minimal model program and the theory of Hodge atoms; further developments shall appear in [6 BGKMM26] , among others.
This paper is organized as follows
In § 2 we recall the basics of quantum cohomology and fix our conventions. Further technicalities used in subsequent sections are also discussed. In particular, the irrationality criteria used here are recalled. In § 3 we compute the vertical product, extend its decomposition, and prove Theorem B. In § 4 we prove Theorem A and Corollary C. Finally, in § 5 we prove Corollary D and give the explicit criterion on the nodes.
AI methodology
The argument for Lemma 3.5 was developed through a series of interactions with ChatGPT 6. This argument (along with the rest of the text) was written up by the authors.
Acknowledgements
L. F. Cavenaghi and L. Katzarkov are supported by the Simons Foundation, grant SFI-MPS-T-Institutes-00007697, and the Ministry of Education and Science of the Republic of Bulgaria, grant DO1-239/10.12.2024. This work was done while the first-named author was visiting the Simons Center for Geometry and Physics, for which he thanks the hospitality and nice work environment. L. Katzarkov is supported by the Simons Investigators Award (no.003136), NSF FRG grant DMS-2245099, and NSF FRG grant DMS-2255039. D. Pomerleano was partially supported by NSF grant DMS-2604648. The authors thank Yuri Tschinkel for constant support and interest.
Basics of Quantum Cohomology
Here, we recall the basic features of quantum cohomology needed to prove our results. Let X X X
N 1 ( X , Z ) : = im ( CH 1 ( X ) → c l H 2 ( X , Q ) ) N_1(X,\mathbb{Z}) := \operatorname{im}\left(\operatorname{CH}_1(X) \xrightarrow{\mathrm{cl}} H_2(X,\mathbb{Q})\right) N 1 ( X , Z ) := im ( CH 1 ( X ) cl H 2 ( X , Q ) ) the subgroup of H 2 ( X , Q ) H_2(X,\mathbb{Q}) H 2 ( X , Q ) 1 1 1 NE ( X , Z ) ⊂ N 1 ( X , Z ) \operatorname{NE}(X,\mathbb{Z}) \subset N_1(X,\mathbb{Z}) NE ( X , Z ) ⊂ N 1 ( X , Z ) X X X H X H_X H X X X X β ∈ NE ( X , Z ) \beta\in\operatorname{NE}(X,\mathbb{Z}) β ∈ NE ( X , Z ) ∫ β H X ≤ k \int_\beta H_X \le k ∫ β H X ≤ k β ∈ NE ( X , Z ) \beta\in\operatorname{NE}(X,\mathbb{Z}) β ∈ NE ( X , Z ) β = β 1 + β 2 \beta= \beta_1 + \beta_2 β = β 1 + β 2 β 1 , β 2 ∈ NE ( X , Z ) \beta_1,\beta_2 \in\operatorname{NE}(X,\mathbb{Z}) β 1 , β 2 ∈ NE ( X , Z ) [15 Lemmas 19–20] . We may therefore define the Novikov ring N X \mathcal{N}_X N X
N X : = Q [ [ NE ( X , Z ) ] ] = { ∑ β ∈ NE ( X , Z ) a β q β : a β ∈ Q } , q β q β ′ = q β + β ′ , \mathcal{N}_X := \mathbb{Q}[[\operatorname{NE}(X,\mathbb{Z})]] = \left\{ \sum_{\beta\in\operatorname{NE}(X,\mathbb{Z})} a_\beta q^\beta: a_\beta\in\mathbb{Q} \right\}, \qquad q^\beta q^{\beta'} = q^{\beta+\beta'}, N X := Q [[ NE ( X , Z )]] = ⎩ ⎨ ⎧ β ∈ NE ( X , Z ) ∑ a β q β : a β ∈ Q ⎭ ⎬ ⎫ , q β q β ′ = q β + β ′ , endowed with the complete separated linear topology defined by the degree filtration
I k : = { ∑ β a β q β : a β = 0 whenever ∫ β H X < k } , k ≥ 1. I_k := \left\{ \sum_\beta a_\beta q^\beta: a_\beta= 0 \text{ whenever } \int_\beta H_X < k \right\}, \qquad k \ge1. I k := ⎩ ⎨ ⎧ β ∑ a β q β : a β = 0 whenever ∫ β H X < k ⎭ ⎬ ⎫ , k ≥ 1. Since N 1 ( X , Z ) N_1(X,\mathbb{Z}) N 1 ( X , Z ) N X \mathcal{N}_X N X
Here and throughout, let H ∗ ( X , Q ) H^*(X,\mathbb{Q}) H ∗ ( X , Q ) X X X
H e v e n ( X , Q ) : = ⨁ k ≥ 0 H 2 k ( X , Q ) . H^{\mathrm{even}}(X,\mathbb{Q}) := \bigoplus_{k\geq0} H^{2k}(X,\mathbb{Q}). H even ( X , Q ) := k ≥ 0 ⨁ H 2 k ( X , Q ) . Write ρ : = rank NS ( X ) \rho:= \operatorname{rank}\operatorname{NS}(X) ρ := rank NS ( X ) X X X Q \mathbb{Q} Q
{ T 0 = 1 , T 1 , T 2 , … , T ρ , T ρ + 1 , … , T m } \{T_0=1,T_1,T_2,\ldots,T_\rho,T_{\rho+1},\ldots,T_m\} { T 0 = 1 , T 1 , T 2 , … , T ρ , T ρ + 1 , … , T m } of H e v e n ( X , Q ) H^{\mathrm{even}}(X,\mathbb{Q}) H even ( X , Q ) T 1 , … , T ρ ∈ H 2 ( X , Q ) T_1,\ldots,T_\rho\in H^2(X,\mathbb{Q}) T 1 , … , T ρ ∈ H 2 ( X , Q ) Z \mathbb{Z} Z NS ( X ) / tors \operatorname{NS}(X)/\operatorname{tors} NS ( X ) / tors Q \mathbb{Q} Q
im ( c 1 : NS ( X ) ⊗ Z Q ⟶ H 2 ( X , Q ) ) = H 2 ( X , Q ) ∩ H 1 , 1 ( X ) , \operatorname{im}\left(c_1:\operatorname{NS}(X)\otimes_{\mathbb{Z}}\mathbb{Q}\longrightarrow H^2(X,\mathbb{Q})\right)=H^2(X,\mathbb{Q})\cap H^{1,1}(X), im ( c 1 : NS ( X ) ⊗ Z Q ⟶ H 2 ( X , Q ) ) = H 2 ( X , Q ) ∩ H 1 , 1 ( X ) , where the intersection is taken inside H 2 ( X , C ) H^2(X,\mathbb{C}) H 2 ( X , C ) H 2 ( X , Q ) → H 2 ( X , C ) H^2(X,\mathbb{Q})\to H^2(X,\mathbb{C}) H 2 ( X , Q ) → H 2 ( X , C )
N 1 ( X , Z ) ⟶ Z ρ , β ⟼ ( ∫ β T 1 , … , ∫ β T ρ ) , N_1(X,\mathbb{Z})\longrightarrow\mathbb{Z}^\rho,\qquad\beta\longmapsto\left(\int_\beta T_1,\ldots,\int_\beta T_\rho\right), N 1 ( X , Z ) ⟶ Z ρ , β ⟼ ( ∫ β T 1 , … , ∫ β T ρ ) , so that we use the Laurent-monomial notation
q β = ∏ i = 1 ρ q i β ⋅ T i . q^\beta=\prod_{i=1}^{\rho}q_i^{\beta\cdot T_i}. q β = i = 1 ∏ ρ q i β ⋅ T i . One remarks, however, that the symbols q i q_i q i N X \mathcal{N}_X N X
Let t 0 , … , t m t_0,\ldots,t_m t 0 , … , t m T 0 , … , T m T_0,\ldots,T_m T 0 , … , T m reduced big Novikov ring by
Nov X r e d : = N X [ [ ( t i ) i ∉ { 1 , … , ρ } ] ] , \operatorname{Nov}^{\mathrm{red}}_X:=\mathcal{N}_X\left[\left[(t_i)_{i\notin\{1,\ldots,\rho\}}\right]\right], Nov X red := N X [ [ ( t i ) i ∈ / { 1 , … , ρ } ] ] , and set
t ′ : = ∑ i ∉ { 1 , … , ρ } t i T i ∈ H e v e n ( X , Q ) ⊗ Q Nov X r e d . t':=\sum_{i\notin\{1,\ldots,\rho\}}t_iT_i\in H^{\mathrm{even}}(X,\mathbb{Q})\otimes_{\mathbb{Q}}\operatorname{Nov}^{\mathrm{red}}_X. t ′ := i ∈ / { 1 , … , ρ } ∑ t i T i ∈ H even ( X , Q ) ⊗ Q Nov X red . The (even)2 big quantum cohomology ring , denoted by Q H e v e n ( X ) QH^{\mathrm{even}}(X) Q H even ( X ) Nov X r e d \operatorname{Nov}^{\mathrm{red}}_X Nov X red
H e v e n ( X , Q ) ⊗ Q Nov X r e d H^{\mathrm{even}}(X,\mathbb{Q})\otimes_{\mathbb{Q}}\operatorname{Nov}^{\mathrm{red}}_X H even ( X , Q ) ⊗ Q Nov X red equipped with the big quantum product ⋆ t ′ \star_{t'} ⋆ t ′ Nov X r e d \operatorname{Nov}^{\mathrm{red}}_X Nov X red γ 1 , γ 2 , γ 3 ∈ H e v e n ( X , Q ) \gamma_1,\gamma_2,\gamma_3\in H^{\mathrm{even}}(X,\mathbb{Q}) γ 1 , γ 2 , γ 3 ∈ H even ( X , Q )
( γ 1 ⋆ t ′ γ 2 , γ 3 ) X : = ∑ β ∈ NE ( X , Z ) ∑ n ≥ 0 1 n ! ∏ i = 1 ρ q i β ⋅ T i ⟨ γ 1 , γ 2 , γ 3 , t ′ , … , t ′ ⏟ n t i m e s ⟩ 0 , n + 3 , β . (\gamma_1\star_{t'}\gamma_2,\gamma_3)_X:=\sum_{\beta\in\operatorname{NE}(X,\mathbb{Z})}\sum_{n\geq0}\frac{1}{n!}\prod_{i=1}^{\rho}q_i^{\beta\cdot T_i}\left\langle\gamma_1,\gamma_2,\gamma_3,\underbrace{t',\ldots,t'}_{n\ \mathrm{times}}\right\rangle_{0,n+3,\beta}. ( γ 1 ⋆ t ′ γ 2 , γ 3 ) X := β ∈ NE ( X , Z ) ∑ n ≥ 0 ∑ n ! 1 i = 1 ∏ ρ q i β ⋅ T i ⟨ γ 1 , γ 2 , γ 3 , n times t ′ , … , t ′ ⟩ 0 , n + 3 , β . Here ( α , γ ) X : = ∫ X α ⌣ γ (\alpha,\gamma)_X:=\int_X\alpha\smile\gamma ( α , γ ) X := ∫ X α ⌣ γ Nov X r e d \operatorname{Nov}^{\mathrm{red}}_X Nov X red π Q : H 2 ( X , Z ) → H 2 ( X , Q ) \pi_{\mathbb{Q}}:H_2(X,\mathbb{Z})\to H_2(X,\mathbb{Q}) π Q : H 2 ( X , Z ) → H 2 ( X , Q )
⟨ α 1 , … , α N ⟩ 0 , N , β : = ∑ β ~ ∈ H 2 ( X , Z ) π Q ( β ~ ) = β ∫ [ M ‾ 0 , N ( X , β ~ ) ] v i r ∏ j = 1 N ev j ∗ ( α j ) , \left\langle\alpha_1,\ldots,\alpha_N\right\rangle_{0,N,\beta}:=\sum_{\substack{\widetilde{\beta}\in H_2(X,\mathbb{Z})\\ \pi_{\mathbb{Q}}(\widetilde{\beta})=\beta}}\int_{[\overline{M}_{0,N}(X,\widetilde{\beta})]^{\mathrm{vir}}}\prod_{j=1}^{N}\operatorname{ev}_j^*(\alpha_j), ⟨ α 1 , … , α N ⟩ 0 , N , β := β ∈ H 2 ( X , Z ) π Q ( β ) = β ∑ ∫ [ M 0 , N ( X , β ) ] vir j = 1 ∏ N ev j ∗ ( α j ) , for N ≥ 0 N\geq0 N ≥ 0 β ≠ 0 \beta\neq0 β = 0 N ≥ 3 N\geq3 N ≥ 3 β = 0 \beta=0 β = 0 Tor H 2 ( X , Z ) \operatorname{Tor}H_2(X,\mathbb{Z}) Tor H 2 ( X , Z ) [8 KM94] for the basic axioms.
Quantum multiplication as a multiplication on the logarithmic tangent bundle
For 1 ≤ i ≤ ρ 1 \le i \le\rho 1 ≤ i ≤ ρ q i ∂ q i q_i\partial_{q_i} q i ∂ q i
q i ∂ q i ( q β ) = ( ∫ β T i ) q β , q i ∂ q i ( t j ) = 0. q_i\partial_{q_i}\left(q^\beta\right)=\left(\int_\beta T_i\right)q^\beta,\qquad q_i\partial_{q_i}(t_j)=0. q i ∂ q i ( q β ) = ( ∫ β T i ) q β , q i ∂ q i ( t j ) = 0. Consider the free module of logarithmic vector fields
T X log : = ⨁ i = 1 ρ Nov X r e d q i ∂ q i ⊕ ⨁ i ∉ { 1 , … , ρ } Nov X r e d ∂ t i . \mathcal{T}^{\log}_X:=\bigoplus_{i=1}^{\rho}\operatorname{Nov}^{\mathrm{red}}_X q_i\partial_{q_i}\oplus\bigoplus_{i\notin\{1,\ldots,\rho\}}\operatorname{Nov}^{\mathrm{red}}_X\partial_{t_i}. T X l o g := i = 1 ⨁ ρ Nov X red q i ∂ q i ⊕ i ∈ / { 1 , … , ρ } ⨁ Nov X red ∂ t i . We identify it with H e v e n ( X , Q ) ⊗ Q Nov X r e d H^{\mathrm{even}}(X,\mathbb{Q})\otimes_{\mathbb{Q}}\operatorname{Nov}^{\mathrm{red}}_X H even ( X , Q ) ⊗ Q Nov X red
q i ∂ q i ⟷ T i ( 1 ≤ i ≤ ρ ) , ∂ t i ⟷ T i ( i ∉ { 1 , … , ρ } ) . q_i\partial_{q_i}\longleftrightarrow T_i\quad(1\le i\le\rho),\qquad\partial_{t_i}\longleftrightarrow T_i\quad(i\notin\{1,\ldots,\rho\}). q i ∂ q i ⟷ T i ( 1 ≤ i ≤ ρ ) , ∂ t i ⟷ T i ( i ∈ / { 1 , … , ρ }) . Write d i : = deg T i d_i:=\deg T_i d i := deg T i T i T_i T i
deg q β : = 2 ∫ β c 1 ( T X ) , deg t i : = 2 − d i . \deg q^\beta:=2\int_\beta c_1(T_X),\qquad\deg t_i:=2-d_i. deg q β := 2 ∫ β c 1 ( T X ) , deg t i := 2 − d i . The dimension axiom implies that ⋆ t ′ \star_{t'} ⋆ t ′
T i ⋆ t ′ T j = ∑ k C i j k T k , T_i\star_{t'}T_j=\sum_k C^k_{ij}T_k, T i ⋆ t ′ T j = k ∑ C ij k T k , then every monomial occurring in C i j k C^k_{ij} C ij k d i + d j − d k d_i+d_j-d_k d i + d j − d k
Eu t ′ : = c 1 ( T X ) + ∑ i ∉ { 1 , … , ρ } ( 1 − 1 2 deg T i ) t i T i ∈ Q H e v e n ( X ) , \operatorname{Eu}_{t'}:=c_1(T_X)+\sum_{i\notin\{1,\ldots,\rho\}}\left(1-\frac{1}{2}\deg T_i\right)t_iT_i\in QH^{\mathrm{even}}(X), Eu t ′ := c 1 ( T X ) + i ∈ / { 1 , … , ρ } ∑ ( 1 − 2 1 deg T i ) t i T i ∈ Q H even ( X ) , upon writing c 1 ( T X ) = ∑ i = 1 ρ r i T i c_1(T_X)=\sum_{i=1}^{\rho}r_iT_i c 1 ( T X ) = ∑ i = 1 ρ r i T i r i ∈ Z r_i\in\mathbb{Z} r i ∈ Z
∑ i = 1 ρ r i q i ∂ q i + ∑ i ∉ { 1 , … , ρ } ( 1 − 1 2 deg T i ) t i ∂ t i . \sum_{i=1}^{\rho}r_iq_i\partial_{q_i}+\sum_{i\notin\{1,\ldots,\rho\}}\left(1-\frac{1}{2}\deg T_i\right)t_i\partial_{t_i}. i = 1 ∑ ρ r i q i ∂ q i + i ∈ / { 1 , … , ρ } ∑ ( 1 − 2 1 deg T i ) t i ∂ t i . Denoting by Lie \operatorname{Lie} Lie ⋆ t ′ \star_{t'} ⋆ t ′
Lie Eu t ′ ( ⋆ t ′ ) = ⋆ t ′ . \operatorname{Lie}_{\operatorname{Eu}_{t'}}(\star_{t'})=\star_{t'}. Lie Eu t ′ ( ⋆ t ′ ) = ⋆ t ′ . Throughout, we work over the algebraic closure K X : = Frac ( Nov X r e d ) ‾ K_X:=\overline{\operatorname{Frac}(\operatorname{Nov}^{\mathrm{red}}_X)} K X := Frac ( Nov X red ) Nov X r e d \operatorname{Nov}^{\mathrm{red}}_X Nov X red H e v e n ( X , K X ) : = H e v e n ( X , Q ) ⊗ Q K X H^{\mathrm{even}}(X,K_X):=H^{\mathrm{even}}(X,\mathbb{Q})\otimes_{\mathbb{Q}}K_X H even ( X , K X ) := H even ( X , Q ) ⊗ Q K X K X K_X K X
κ t ′ : = ( Eu t ′ ⋆ t ′ ( − ) ) ∈ End K X ( H e v e n ( X , K X ) ) . \kappa_{t'}:=\left(\operatorname{Eu}_{t'}\star_{t'}(-)\right)\in\operatorname{End}_{K_X}\left(H^{\mathrm{even}}(X,K_X)\right). κ t ′ := ( Eu t ′ ⋆ t ′ ( − ) ) ∈ End K X ( H even ( X , K X ) ) . By the multiplicity of an eigenvalue of κ t ′ \kappa_{t'} κ t ′ K X K_X K X κ t ′ \kappa_{t'} κ t ′
Remark 2.1. The construction above is the even, reduced formal version of the base of the A-model F-bundle of [13 Sections 3.5.1–3.5.2] .
The main rationality obstruction we shall use comes from the theory of Hodge atoms of Katzarkov–Kontsevich–Pantev–Yu in the following simplified form observed by Cai [5 ] :
Proposition 2.2 ([5 Theorem 5.6] , after [13 ] ). For every smooth projective rational threefold X X X κ t ′ \kappa_{t'} κ t ′
Remark 2.3. All completions used in this paper are ordinary completions. The proof of Theorem 2.2, however, relies on the blowup formula of [11 ] , which is formulated using graded completions; see [11 Remark 1.3 and Section 2.2] . This distinction causes no difficulty here. By the base-change comparison of § 2.3 , the algebraic multiplicities of the eigenvalues of the Euler multiplication operator are preserved after passing between the corresponding coefficient fields.
Local Artinian factors
We shall use the algebra structure of quantum cohomology to obtain lower bounds for the algebraic multiplicities of the eigenvalues of κ t ′ \kappa_{t}' κ t ′
Of interest to us is the finite-dimensional commutative unital K X \mathbb{K}_X K X
A X : = Q H e v e n ( X ) ⊗ Nov X r e d K X . A_X := QH^{\mathrm{even}}(X) \otimes\operatorname{Nov}_{X}^{\mathrm{red}} \mathbb{K}_X. A X := Q H even ( X ) ⊗ Nov X red K X . As a finite-dimensional commutative algebra, A X A_X A X
A X ≅ ∏ ν = 1 s A ν . A_X \cong\prod_{\nu=1}^{s} A_\nu. A X ≅ ν = 1 ∏ s A ν . where the A ν A_\nu A ν K X \mathbb{K}_X K X local factors of A X A_X A X
Let m ν \mathfrak{m}_\nu m ν A ν A_\nu A ν K X \mathbb{K}_X K X
A ν / m ν ≅ K X . A_\nu/\mathfrak{m}_\nu\cong\mathbb{K}_X. A ν / m ν ≅ K X . Write the image of E u t ′ \mathrm{Eu}_{t}' Eu t ′ A ν A_\nu A ν
E u ν = λ ν 1 ν + n ν , λ ν ∈ K X , n ν ∈ m ν , \mathrm{Eu}_\nu=\lambda_\nu1_\nu+n_\nu,\qquad\lambda_\nu\in\mathbb{K}_X,\qquad n_\nu\in\mathfrak{m}_\nu, Eu ν = λ ν 1 ν + n ν , λ ν ∈ K X , n ν ∈ m ν , where 1 ν 1_\nu 1 ν A ν A_\nu A ν A ν A_\nu A ν A X A_X A X
κ t ′ ∣ A ν = λ ν id A ν + L n ν . \left.\kappa_{t}'\right|_{A_\nu}=\lambda_\nu\operatorname{id}_{A_\nu}+L_{n_\nu}. κ t ′ ∣ A ν = λ ν id A ν + L n ν . Here L n ν L_{n_\nu} L n ν n ν n_\nu n ν m ν \mathfrak{m}_\nu m ν
det ( z id A X − κ t ′ ) = ∏ ν = 1 s ( z − λ ν ) dim K X A ν . \det\left(z\operatorname{id}_{A_X}-\kappa_{t}'\right)=\prod_{\nu=1}^{s}(z-\lambda_\nu)^{\dim_{\mathbb{K}_X}A_\nu}. det ( z id A X − κ t ′ ) = ν = 1 ∏ s ( z − λ ν ) d i m K X A ν . More precisely, the generalized eigenspace for an eigenvalue λ \lambda λ
⨁ λ ν = λ A ν . \bigoplus_{\lambda_\nu=\lambda} A_\nu. λ ν = λ ⨁ A ν . Indeed, κ t ′ − λ id \kappa_{t}'-\lambda\operatorname{id} κ t ′ − λ id λ ν = λ \lambda_\nu=\lambda λ ν = λ
In particular, a local factor of dimension d d d d d d Proposition 2.2 , a local factor of dimension at least three obstructs rationality for a smooth projective threefold.
The arguments in § 3 require changes of coefficient rings. We conclude this section by remarking that the injective changes considered below preserve the dimensions of local factors and their radical powers, as well as the eigenvalue multiplicities of κ t ′ \kappa_{t}' κ t ′
Changing coefficient rings
Let
Nov X ∘ : = Q [ C i j k , ( κ t ′ ) i j ] ⊂ Nov X r e d , \operatorname{Nov}_{X}^{\circ}:=\mathbb{Q}\left[C_{ij}^{k},(\kappa_{t}')_{ij}\right]\subset\operatorname{Nov}_{X}^{\mathrm{red}}, Nov X ∘ := Q [ C ij k , ( κ t ′ ) ij ] ⊂ Nov X red , where as above the entries C i j k C_{ij}^{k} C ij k ι : Nov X ∘ ↪ O \iota:\operatorname{Nov}_{X}^{\circ}\hookrightarrow\mathcal{O} ι : Nov X ∘ ↪ O ι \iota ι Frac ( O ) \operatorname{Frac}(\mathcal{O}) Frac ( O ) K X K_X K X
Indeed, after choosing compatible field embeddings, both constructions are obtained from the same algebra and operator over Frac ( N o v X ∘ ) \operatorname{Frac}(\mathrm{Nov}^{\circ}_X) Frac ( Nov X ∘ )
Quantum cohomology of standard conic bundles
Throughout this section, let π : X → S \pi: X \to S π : X → S S S S Δ ⊂ S \Delta\subset S Δ ⊂ S X X X π \pi π ρ ( X / S ) = 1 \rho(X/S) = 1 ρ ( X / S ) = 1
Δ S : = 2 K S + Δ , ρ S : = rank NS ( S ) = b 2 ( S ) . \Delta_S := 2K_S + \Delta,\qquad\rho_S := \operatorname{rank}\operatorname{NS}(S) = b_2(S). Δ S := 2 K S + Δ , ρ S := rank NS ( S ) = b 2 ( S ) . Then ρ : = rank NS ( X ) = ρ S + 1 \rho:= \operatorname{rank}\operatorname{NS}(X) = \rho_S + 1 ρ := rank NS ( X ) = ρ S + 1
Theorem 3.1. Let π : X → S \pi: X \to S π : X → S S S S Δ S = 2 K S + Δ \Delta_S = 2K_S + \Delta Δ S = 2 K S + Δ
Q H e v e n ( X ) ⊗ N o v X r e d K X QH^{\mathrm{even}}(X) \otimes\mathrm{Nov}^{\mathrm{red}}_X K_X Q H even ( X ) ⊗ Nov X red K X has a local Artinian factor H − \mathcal{H}_- H −
dim K X H − = ρ S + 2 , dim K X rad ( H − ) = ρ S + 1 , \dim_{K_X}\mathcal{H}_- = \rho_S + 2,\qquad\dim_{K_X}\operatorname{rad}(\mathcal{H}_-) = \rho_S + 1, dim K X H − = ρ S + 2 , dim K X rad ( H − ) = ρ S + 1 , dim K X ( rad ( H − ) 2 ) = 1 , rad ( H − ) 3 = 0. \dim_{K_X}\left(\operatorname{rad}(\mathcal{H}_-)^2\right) = 1,\qquad\operatorname{rad}(\mathcal{H}_-)^3 = 0. dim K X ( rad ( H − ) 2 ) = 1 , rad ( H − ) 3 = 0. Moreover, κ t ′ \kappa_t' κ t ′ ρ S + 2 ≥ 3 \rho_S+2 \ge3 ρ S + 2 ≥ 3 X X X
Classical cohomology
Since the fibers of π \pi π
π ∗ O X = O S , R j π ∗ O X = 0 ( j > 0 ) . \pi_*\mathcal{O}_X = \mathcal{O}_S,\qquad R^j\pi_*\mathcal{O}_X = 0\quad(j > 0). π ∗ O X = O S , R j π ∗ O X = 0 ( j > 0 ) . As S S S H 2 ( X , O X ) = 0 H^2(X,\mathcal{O}_X) = 0 H 2 ( X , O X ) = 0
b 2 ( X ) = b 4 ( X ) = ρ S + 1 , dim Q H e v e n ( X , Q ) = 2 ρ S + 4. b_2(X) = b_4(X) = \rho_S + 1,\qquad\dim_{\mathbb{Q}} H^{\mathrm{even}}(X,\mathbb{Q}) = 2\rho_S + 4. b 2 ( X ) = b 4 ( X ) = ρ S + 1 , dim Q H even ( X , Q ) = 2 ρ S + 4. Let ℓ ∈ N 1 ( X , Z ) \ell\in N_1(X,\mathbb{Z}) ℓ ∈ N 1 ( X , Z ) Δ \Delta Δ
∫ ℓ c 1 ( T X ) = 1. \int_{\ell} c_1(T_X) = 1. ∫ ℓ c 1 ( T X ) = 1. Since ρ ( X / S ) = 1 \rho(X/S) = 1 ρ ( X / S ) = 1 2 ℓ 2\ell 2 ℓ
ker ( π ∗ : N 1 ( X , Z ) → N 1 ( S , Z ) ) = Z ℓ . \ker\left(\pi_* : N_1(X,\mathbb{Z}) \to N_1(S,\mathbb{Z})\right) = \mathbb{Z}\ell. ker ( π ∗ : N 1 ( X , Z ) → N 1 ( S , Z ) ) = Z ℓ . Indeed, the rational kernel is Q ℓ \mathbb{Q}\ell Q ℓ a ℓ ∈ N 1 ( X , Z ) a\ell\in N_1(X,\mathbb{Z}) a ℓ ∈ N 1 ( X , Z ) a = ∫ a ℓ c 1 ( T X ) ∈ Z a = \int_{a\ell} c_1(T_X) \in\mathbb{Z} a = ∫ a ℓ c 1 ( T X ) ∈ Z
Write
H S : = H e v e n ( S , Q ) = Q 1 ⊕ D S ⊕ Q p , D S : = H 2 ( S , Q ) , H_S := H^{\mathrm{even}}(S,\mathbb{Q}) = \mathbb{Q}1 \oplus D_S \oplus\mathbb{Q}p,\qquad D_S := H^2(S,\mathbb{Q}), H S := H even ( S , Q ) = Q 1 ⊕ D S ⊕ Q p , D S := H 2 ( S , Q ) , where p = [ p t ] p = [\mathrm{pt}] p = [ pt ] ∫ S p = 1 \int_S p = 1 ∫ S p = 1 Q \mathbb{Q} Q Δ 1 , … , Δ ρ S \Delta_1,\ldots,\Delta_{\rho_S} Δ 1 , … , Δ ρ S D S D_S D S
g i j : = ∫ S Δ i ⌣ Δ j . g_{ij} := \int_S \Delta_i \smile\Delta_j. g ij := ∫ S Δ i ⌣ Δ j . Then ( g i j ) (g_{ij}) ( g ij )
Δ i ⌣ Δ j = g i j p , Δ i ⌣ p = p ⌣ 2 = 0. \Delta_i \smile\Delta_j = g_{ij}p,\qquad\Delta_i \smile p = p^{\smile2} = 0. Δ i ⌣ Δ j = g ij p , Δ i ⌣ p = p ⌣ 2 = 0. Hereafter, we use the same notation for a class on S S S X X X
Let
ω X / S : = ω X ⊗ π ∗ ω S − 1 , h : = c 1 ( ω X / S − 1 ) = c 1 ( T X ) − π ∗ c 1 ( T S ) . \omega_{X/S}:=\omega_X\otimes\pi^*\omega_S^{-1},\qquad h:=c_1(\omega_{X/S}^{-1})=c_1(T_X)-\pi^*c_1(T_S). ω X / S := ω X ⊗ π ∗ ω S − 1 , h := c 1 ( ω X / S − 1 ) = c 1 ( T X ) − π ∗ c 1 ( T S ) . Since h h h π ∗ h = 2 \pi_*h=2 π ∗ h = 2 [16 Lemma 3.11] ,
π ∗ ( K X 2 ) ≡ − 4 K S − Δ . \pi_*(K_X^2)\equiv-4K_S-\Delta. π ∗ ( K X 2 ) ≡ − 4 K S − Δ. Because numerically equivalent divisors on S S S H 2 ( S , Q ) H^2(S,\mathbb{Q}) H 2 ( S , Q ) h = − K X + π ∗ K S h=-K_X+\pi^*K_S h = − K X + π ∗ K S π ∗ K X = − 2 \pi_*K_X=-2 π ∗ K X = − 2
π ∗ ( h 2 ) = π ∗ ( K X 2 ) − 2 K S π ∗ K X = − Δ . \pi_*(h^2)=\pi_*(K_X^2)-2K_S\pi_*K_X=-\Delta. π ∗ ( h 2 ) = π ∗ ( K X 2 ) − 2 K S π ∗ K X = − Δ. Define
x : = 1 2 h + 1 8 Δ . x:=\frac{1}{2}h+\frac{1}{8}\Delta. x := 2 1 h + 8 1 Δ. The projection formula gives
π ∗ x = 1 , π ∗ ( x 2 ) = 1 4 π ∗ ( h 2 ) + 1 8 Δ π ∗ h = 0. \pi_*x=1,\qquad\pi_*(x^2)=\frac{1}{4}\pi_*(h^2)+\frac{1}{8}\Delta\pi_*h=0. π ∗ x = 1 , π ∗ ( x 2 ) = 4 1 π ∗ ( h 2 ) + 8 1 Δ π ∗ h = 0. These identities give the following presentation.
Lemma 3.2. There is an isomorphism of graded H S H_S H S
H e v e n ( X , Q ) = π ∗ H S ⊕ ( x ⌣ π ∗ H S ) . H^{\mathrm{even}}(X,\mathbb{Q})=\pi^*H_S\oplus\left(x\smile\pi^*H_S\right). H even ( X , Q ) = π ∗ H S ⊕ ( x ⌣ π ∗ H S ) . Moreover, there is a unique α ∈ Q \alpha\in\mathbb{Q} α ∈ Q x 2 = α p x^2=\alpha p x 2 = α p a , b , a 1 , a 2 , b 1 , b 2 ∈ H S a,b,a_1,a_2,b_1,b_2\in H_S a , b , a 1 , a 2 , b 1 , b 2 ∈ H S
∫ X ( a + x b ) = ∫ S b , c 1 ( T X ) = 2 x + c 1 ( T S ) − 1 4 Δ , \int_X(a+xb)=\int_S b,\qquad c_1(T_X)=2x+c_1(T_S)-\frac{1}{4}\Delta, ∫ X ( a + x b ) = ∫ S b , c 1 ( T X ) = 2 x + c 1 ( T S ) − 4 1 Δ , and
( a 1 + x b 1 , a 2 + x b 2 ) X = ∫ S ( a 1 b 2 + b 1 a 2 ) . (a_1+xb_1,a_2+xb_2)_X=\int_S(a_1b_2+b_1a_2). ( a 1 + x b 1 , a 2 + x b 2 ) X = ∫ S ( a 1 b 2 + b 1 a 2 ) . Here, integration of an inhomogeneous class means integration of its top-degree component.
Proof. Since π ∗ ( x π ∗ a ) = a \pi_*(x\pi^*a)=a π ∗ ( x π ∗ a ) = a π ∗ \pi^* π ∗ π ∗ a + x π ∗ b = 0 \pi^*a+x\pi^*b=0 π ∗ a + x π ∗ b = 0 π ∗ \pi_* π ∗ b = 0 b=0 b = 0 a = 0 a=0 a = 0
π ∗ H S ⊕ x π ∗ H S ⟶ H e v e n ( X , Q ) \pi^*H_S\oplus x\pi^*H_S\longrightarrow H^{\mathrm{even}}(X,\mathbb{Q}) π ∗ H S ⊕ x π ∗ H S ⟶ H even ( X , Q ) is an isomorphism because both sides have dimension 2 ρ S + 4 2\rho_S+4 2 ρ S + 4 π ∗ \pi_* π ∗ Q p \mathbb{Q}p Q p π ∗ ( x 2 ) = 0 \pi_*(x^2)=0 π ∗ ( x 2 ) = 0 x 2 = α p x^2=\alpha p x 2 = α p α ∈ Q \alpha\in\mathbb{Q} α ∈ Q π ∗ x = 1 \pi_*x=1 π ∗ x = 1 π ∗ ( x 2 ) = 0 \pi_*(x^2)=0 π ∗ ( x 2 ) = 0 c 1 ( T X ) c_1(T_X) c 1 ( T X ) x x x □ \square □
We now choose coordinates adapted to this presentation. Since c 1 c_1 c 1 NS ( X ) ⊗ Q \operatorname{NS}(X)\otimes\mathbb{Q} NS ( X ) ⊗ Q H 2 ( X , Q ) H^2(X,\mathbb{Q}) H 2 ( X , Q )
1 , p , x Δ 1 , … , x Δ ρ S , x p . 1,\quad p,\quad x\Delta_1,\ldots,x\Delta_{\rho_S},\quad xp. 1 , p , x Δ 1 , … , x Δ ρ S , x p . Let u , w , c 1 , … , c ρ S , ξ u,w,c^1,\ldots,c^{\rho_S},\xi u , w , c 1 , … , c ρ S , ξ
t ′ = a ′ + x b ′ , (2) t'=a'+xb', \tag*{(2)} t ′ = a ′ + x b ′ , (2) where
a ′ : = u 1 + w p , b ′ : = c + ξ p , c : = ∑ i = 1 ρ S c i Δ i . a':=u1+wp,\qquad b':=c+\xi p,\qquad c:=\sum_{i=1}^{\rho_S}c^i\Delta_i. a ′ := u 1 + w p , b ′ := c + ξ p , c := i = 1 ∑ ρ S c i Δ i . The classes 1 , p , x Δ i , x p 1,p,x\Delta_i,xp 1 , p , x Δ i , x p 0 , 4 , 4 , 6 0,4,4,6 0 , 4 , 4 , 6
deg u = 2 , deg w = deg c i = − 2 , deg ξ = − 4. \deg u=2,\qquad\deg w=\deg c^i=-2,\qquad\deg\xi=-4. deg u = 2 , deg w = deg c i = − 2 , deg ξ = − 4. Moreover,
E u t ′ = 2 x + c 1 ( T S ) − 1 4 Δ + u 1 − w p − x c − 2 ξ x p . (3) E u_{t'} = 2x + c_1(T_S) - \frac{1}{4}\Delta+ u1 - wp - xc - 2\xi xp. \tag*{(3)} E u t ′ = 2 x + c 1 ( T S ) − 4 1 Δ + u 1 − w p − x c − 2 ξ x p . (3) The vertical quantum product
A class β ∈ NE ( X , Z ) \beta\in\operatorname{NE}(X,\mathbb{Z}) β ∈ NE ( X , Z ) vertical if π ∗ β = 0 \pi_*\beta= 0 π ∗ β = 0 β = m ℓ \beta= m\ell β = m ℓ m ≥ 0 m \ge0 m ≥ 0 q f : = q ℓ q_f := q^\ell q f := q ℓ vertical quantum product ⋆ v e r t \star_{\mathrm{vert}} ⋆ vert ⋆ t ′ \star_{t'} ⋆ t ′ q β = 0 q^\beta= 0 q β = 0
We compute this product and show that, after changing coefficients, its algebra splits into two copies of H S H_S H S H S H_S H S ( b ′ ) 3 = 0 (b')^3 = 0 ( b ′ ) 3 = 0
e b ′ = 1 + b ′ + 1 2 ( b ′ ) 2 , e ± b ′ / 2 = 1 ± 1 2 b ′ + 1 8 ( b ′ ) 2 . e^{b'} = 1 + b' + \frac{1}{2}(b')^2,\qquad e^{\pm b'/2} = 1 \pm\frac{1}{2}b' + \frac{1}{8}(b')^2. e b ′ = 1 + b ′ + 2 1 ( b ′ ) 2 , e ± b ′ /2 = 1 ± 2 1 b ′ + 8 1 ( b ′ ) 2 . Proposition 3.3. (i) By multilinearity and the fundamental-class and divisor axioms, the positive-degree vertical primary genus-zero invariants with even insertions are determined by
⟨ x D ⟩ 0 , 1 , ℓ = ∫ S Δ D , ⟨ x p ⟩ 0 , 1 , 2 ℓ = 1 , ⟨ x D 1 , x D 2 ⟩ 0 , 2 , 2 ℓ = ∫ S D 1 D 2 , (4) \begin{aligned}
\langle xD\rangle_{0,1,\ell} &= \int_S \Delta D,\\
\langle xp\rangle_{0,1,2\ell} &= 1,\\
\langle xD_1,xD_2\rangle_{0,2,2\ell} &= \int_S D_1D_2, \tag*{(4)}
\end{aligned} ⟨ x D ⟩ 0 , 1 , ℓ ⟨ x p ⟩ 0 , 1 , 2 ℓ ⟨ x D 1 , x D 2 ⟩ 0 , 2 , 2 ℓ = ∫ S Δ D , = 1 , = ∫ S D 1 D 2 , (4) D , D 1 , D 2 ∈ D S D,D_1,D_2 \in D_S D , D 1 , D 2 ∈ D S
Among invariants with insertions in { p , x Δ i , x p } \{p,x\Delta_i,xp\} { p , x Δ i , x p } a 1 , a 2 , b 1 , b 2 ∈ H S a_1,a_2,b_1,b_2 \in H_S a 1 , a 2 , b 1 , b 2 ∈ H S
( a 1 + x b 1 ) ⋆ v e r t ( a 2 + x b 2 ) = a 1 a 2 + B ( b ′ ) b 1 b 2 + x ( a 1 b 2 + a 2 b 1 ) , (a_1+xb_1)\star_{\mathrm{vert}}(a_2+xb_2)=a_1a_2+\mathcal{B}(b')b_1b_2+x(a_1b_2+a_2b_1), ( a 1 + x b 1 ) ⋆ vert ( a 2 + x b 2 ) = a 1 a 2 + B ( b ′ ) b 1 b 2 + x ( a 1 b 2 + a 2 b 1 ) , where
B ( b ′ ) : = x ⋆ v e r t x = α p + q f 2 e b ′ + 1 4 q f Δ e b ′ / 2 . \mathcal{B}(b') := x\star_{\mathrm{vert}}x=\alpha p+q_f^2e^{b'}+\frac{1}{4}q_f\Delta e^{b'/2}. B ( b ′ ) := x ⋆ vert x = α p + q f 2 e b ′ + 4 1 q f Δ e b ′ /2 . (ii) The product in (i) has coefficients in
R 0 : = Q [ q f , u , w , c 1 , … , c ρ S , ξ ] . R_0:=\mathbb{Q}[q_f,u,w,c^1,\ldots,c^{\rho_S},\xi]. R 0 := Q [ q f , u , w , c 1 , … , c ρ S , ξ ] . Regard it as an algebra over R 0 R_0 R 0
R f : = k f [ [ u , w , c 1 , … , c ρ S , ξ ] ] , k f : = Q ( q f ) . R_f:=k_f[[u,w,c^1,\ldots,c^{\rho_S},\xi]],\qquad k_f:=\mathbb{Q}(q_f). R f := k f [[ u , w , c 1 , … , c ρ S , ξ ]] , k f := Q ( q f ) . Set
η : = α p − 1 64 Δ 2 = γ p , γ : = α − 1 64 ∫ S Δ 2 , \eta:=\alpha p-\frac{1}{64}\Delta^2=\gamma p,\qquad\gamma:=\alpha-\frac{1}{64}\int_S\Delta^2, η := α p − 64 1 Δ 2 = γ p , γ := α − 64 1 ∫ S Δ 2 , and define
r ( b ′ ) : = q f e b ′ / 2 + 1 8 Δ + η 2 q f e − b ′ / 2 , r(b'):=q_fe^{b'/2}+\frac{1}{8}\Delta+\frac{\eta}{2q_f}e^{-b'/2}, r ( b ′ ) := q f e b ′ /2 + 8 1 Δ + 2 q f η e − b ′ /2 , R ( b ′ ) : = 2 q f e b ′ / 2 + 1 8 Δ b ′ − η q f e − b ′ / 2 , \mathcal{R}(b'):=2q_fe^{b'/2}+\frac{1}{8}\Delta b'-\frac{\eta}{q_f}e^{-b'/2}, R ( b ′ ) := 2 q f e b ′ /2 + 8 1 Δ b ′ − q f η e − b ′ /2 , t ± : = a ′ ± R ( b ′ ) . t_\pm:=a'\pm\mathcal{R}(b'). t ± := a ′ ± R ( b ′ ) . Then r ( b ′ ) 2 = B ( b ′ ) r(b')^2=\mathcal{B}(b') r ( b ′ ) 2 = B ( b ′ )
Φ ( a + x b ) : = ( a + r ( b ′ ) b , a − r ( b ′ ) b ) \Phi(a+xb):=\left(a+r(b')b,\ a-r(b')b\right) Φ ( a + x b ) := ( a + r ( b ′ ) b , a − r ( b ′ ) b ) is an R f R_f R f
( H S ⊗ Q R f ) × ( H S ⊗ Q R f ) , (H_S\otimes_{\mathbb{Q}}R_f)\times(H_S\otimes_{\mathbb{Q}}R_f), ( H S ⊗ Q R f ) × ( H S ⊗ Q R f ) , where each factor is taken with the classical cup product.
(iii) Let ( E u + , E u − ) : = Φ ( E u t ′ ) (\mathrm{Eu}_{+},\mathrm{Eu}_{-}):=\Phi(\mathrm{Eu}_{t^{\prime}}) ( Eu + , Eu − ) := Φ ( Eu t ′ ) Writing
t ± = ( t ± ) 01 + ( t ± ) 1 + ( t ± ) 2 p , ( t ± ) 1 ∈ D S ⊗ Q R f , t_{\pm}=(t_{\pm})_{01}+(t_{\pm})_{1}+(t_{\pm})_{2}p,\qquad(t_{\pm})_{1}\in D_{S}\otimes_{\mathbb{Q}}R_{f}, t ± = ( t ± ) 01 + ( t ± ) 1 + ( t ± ) 2 p , ( t ± ) 1 ∈ D S ⊗ Q R f , we have
E u + = ( t + ) 01 + c 1 ( T S ) − ( t + ) 2 p , E u − = ( t − ) 01 − 1 2 Δ S − ( t − ) 2 p . (5) \begin{aligned}
\mathrm{Eu}_{+}&=(t_{+})_{01}+c_{1}(T_{S})-(t_{+})_{2}p,\\
\mathrm{Eu}_{-}&=(t_{-})_{01}-\frac{1}{2}\Delta_{S}-(t_{-})_{2}p. \tag*{(5)}
\end{aligned} Eu + Eu − = ( t + ) 01 + c 1 ( T S ) − ( t + ) 2 p , = ( t − ) 01 − 2 1 Δ S − ( t − ) 2 p . (5) Proof. (i) A stable map of class m ℓ m\ell m ℓ m > 0 m>0 m > 0 π \pi π
b m , n : M ‾ 0 , n ( X , m ℓ ) ⟶ S b_{m,n}\colon\overline{M}_{0,n}(X,m\ell)\longrightarrow S b m , n : M 0 , n ( X , m ℓ ) ⟶ S record that fiber.
After applying the fundamental-class and divisor axioms, the remaining insertions are p , x D , x p p,xD,xp p , x D , x p D ∈ D S D\in D_{S} D ∈ D S γ i = x ϵ i π ∗ α i \gamma_{i}=x^{\epsilon_{i}}\pi^{*}\alpha_{i} γ i = x ϵ i π ∗ α i ϵ i ∈ { 0 , 1 } \epsilon_{i}\in\{0,1\} ϵ i ∈ { 0 , 1 } π ∘ e v i = b m , n \pi\circ\mathrm{ev}_{i}=b_{m,n} π ∘ ev i = b m , n b m , n ∗ ( α 1 ⋯ α n ) b_{m,n}^{*}(\alpha_{1}\cdots\alpha_{n}) b m , n ∗ ( α 1 ⋯ α n ) ∑ i deg α i ≤ 4 \sum_{i}\deg\alpha_{i}\leq4 ∑ i deg α i ≤ 4 ( deg γ i , deg α i ) (\deg\gamma_{i},\deg\alpha_{i}) ( deg γ i , deg α i ) ( 4 , 4 ) , ( 4 , 2 ) , ( 6 , 4 ) (4,4),(4,2),(6,4) ( 4 , 4 ) , ( 4 , 2 ) , ( 6 , 4 )
2 m = ∑ i ( deg γ i − 2 ) ≤ ∑ i deg α i ≤ 4. 2m=\sum_{i}(\deg\gamma_{i}-2)\leq\sum_{i}\deg\alpha_{i}\leq4. 2 m = i ∑ ( deg γ i − 2 ) ≤ i ∑ deg α i ≤ 4. Thus, the only possibilities are the three types in (4) and ⟨ p ⟩ 0 , 1 , ℓ \langle p\rangle_{0,1,\ell} ⟨ p ⟩ 0 , 1 , ℓ m > 0 m>0 m > 0
We first compute the class-ℓ \ell ℓ Δ \Delta Δ L L L π \pi π Δ \Delta Δ L L L ( − 1 ) (-1) ( − 1 )
N L / X ≃ O P 1 ( − 1 ) ⊕ O P 1 . N_{L/X}\simeq\mathcal{O}_{\mathbb{P}^{1}}(-1)\oplus\mathcal{O}_{\mathbb{P}^{1}}. N L / X ≃ O P 1 ( − 1 ) ⊕ O P 1 . Hence these maps are unobstructed, and their one-point evaluation map has generic degree one onto π − 1 ( Δ ) \pi^{-1}(\Delta) π − 1 ( Δ ) Sing ( Δ ) \operatorname{Sing}(\Delta) Sing ( Δ ) π − 1 ( Sing ( Δ ) ) \pi^{-1}(\operatorname{Sing}(\Delta)) π − 1 ( Sing ( Δ ))
ev ∗ [ M ‾ 0 , 1 ( X , ℓ ) ] v i r = [ π − 1 ( Δ ) ] . \operatorname{ev}_{*}\left[\overline{M}_{0,1}(X,\ell)\right]^{\mathrm{vir}}=\left[\pi^{-1}(\Delta)\right]. ev ∗ [ M 0 , 1 ( X , ℓ ) ] vir = [ π − 1 ( Δ ) ] . Therefore
⟨ x D ⟩ 0 , 1 , ℓ = ∫ X x D Δ = ∫ S Δ D , ⟨ p ⟩ 0 , 1 , ℓ = ∫ X p Δ = 0. \langle xD\rangle_{0,1,\ell}=\int_{X}xD\Delta=\int_{S}\Delta D,\qquad\langle p\rangle_{0,1,\ell}=\int_{X}p\Delta=0. ⟨ x D ⟩ 0 , 1 , ℓ = ∫ X x D Δ = ∫ S Δ D , ⟨ p ⟩ 0 , 1 , ℓ = ∫ X p Δ = 0. Over S ∖ Δ S\setminus\Delta S ∖ Δ 2 ℓ 2\ell 2 ℓ F ≃ P 1 F\simeq\mathbb{P}^{1} F ≃ P 1 ∫ F x = 1 \int_{F}x=1 ∫ F x = 1 j = 1 , 2 j=1,2 j = 1 , 2 F j F^{j} F j j j j x x x ( ∫ F x ) j = 1 (\int_{F}x)^{j}=1 ( ∫ F x ) j = 1
( b 2 , j ) ∗ ( ∏ i = 1 j ev i ∗ x ∩ [ M ‾ 0 , j ( X , 2 ℓ ) ] v i r ) = [ S ] , j = 1 , 2. (b_{2,j})_{*}\left(\prod_{i=1}^{j}\operatorname{ev}_{i}^{*}x\cap\left[\overline{M}_{0,j}(X,2\ell)\right]^{\mathrm{vir}}\right)=[S],\qquad j=1,2. ( b 2 , j ) ∗ ( i = 1 ∏ j ev i ∗ x ∩ [ M 0 , j ( X , 2 ℓ ) ] vir ) = [ S ] , j = 1 , 2. The equality holds globally because no two-dimensional cycle can be supported on Δ \Delta Δ p p p j = 1 j=1 j = 1 D 1 D 2 D_{1}D_{2} D 1 D 2 (4) .
Divisors pulled back from S S S H S H_{S} H S
a ⋆ v e r t γ = a γ ( a ∈ H S , γ ∈ H e v e n ( X , Q ) ) . a\star_{\mathrm{vert}}\gamma=a\gamma\qquad(a\in H_{S},\ \gamma\in H^{\mathrm{even}}(X,\mathbb{Q})). a ⋆ vert γ = aγ ( a ∈ H S , γ ∈ H even ( X , Q )) . It remains to determine B : = x ⋆ v e r t x B:=x\star_{\mathrm{vert}}x B := x ⋆ vert x a ∈ H S a\in H_{S} a ∈ H S
( B , a ) X = ( x , x a ) X = ∫ X x 2 a = 0. (B,a)_{X}=(x,xa)_{X}=\int_{X}x^{2}a=0. ( B , a ) X = ( x , x a ) X = ∫ X x 2 a = 0. By Lemma 3.2 , this means that the coefficients of B B B H S H_{S} H S
Using the computed invariants and ∫ ℓ x = 1 / 2 \int_{\ell}x=1/2 ∫ ℓ x = 1/2 ∫ 2 ℓ x = 1 \int_{2\ell}x=1 ∫ 2 ℓ x = 1
( B , x p ) X = q f 2 , (\mathcal{B},xp)_{X}=q_{f}^{2}, ( B , x p ) X = q f 2 , ( B , x D ) X = q f 2 ∫ S c D + 1 4 q f ∫ S Δ D , (\mathcal{B},xD)_{X}=q_{f}^{2}\int_{S}cD+\frac{1}{4}q_{f}\int_{S}\Delta D, ( B , x D ) X = q f 2 ∫ S cD + 4 1 q f ∫ S Δ D , ( B , x ) X = α + q f 2 ξ + q f 2 2 ∫ S c 2 + q f 8 ∫ S Δ c . (\mathcal{B},x)_{X}=\alpha+q_{f}^{2}\xi+\frac{q_{f}^{2}}{2}\int_{S}c^{2}+\frac{q_{f}}{8}\int_{S}\Delta c. ( B , x ) X = α + q f 2 ξ + 2 q f 2 ∫ S c 2 + 8 q f ∫ S Δ c . These pairings determine
B = α p + q f 2 e b ′ + 1 4 q f Δ e b ′ / 2 . \mathcal{B}=\alpha p+q_{f}^{2}e^{b'}+\frac{1}{4}q_{f}\Delta e^{b'/2}. B = α p + q f 2 e b ′ + 4 1 q f Δ e b ′ /2 . The general product formula follows from the classical H S H_{S} H S
(ii) The structure constants in (i) belong to R 0 R_{0} R 0 R f R_{f} R f Δ η = η 2 = 0 \Delta\eta=\eta^{2}=0 Δ η = η 2 = 0
r ( b ′ ) 2 = q f 2 e b ′ + 1 4 q f Δ e b ′ / 2 + 1 64 Δ 2 + η = B ( b ′ ) . r(b')^{2}=q_{f}^{2}e^{b'}+\frac{1}{4}q_{f}\Delta e^{b'/2}+\frac{1}{64}\Delta^{2}+\eta=\mathcal{B}(b'). r ( b ′ ) 2 = q f 2 e b ′ + 4 1 q f Δ e b ′ /2 + 64 1 Δ 2 + η = B ( b ′ ) . Moreover, r ( b ′ ) r(b') r ( b ′ ) q f q_{f} q f
The vertical algebra is generated over H S ⊗ R f H_{S}\otimes R_{f} H S ⊗ R f x x x x ⋆ v e r t x = r ( b ′ ) 2 x\star_{\mathrm{vert}}x=r(b')^{2} x ⋆ vert x = r ( b ′ ) 2 x = ± r ( b ′ ) x=\pm r(b') x = ± r ( b ′ ) Φ \Phi Φ
( A + , A − ) ⟼ A + + A − 2 + x A + − A − 2 r ( b ′ ) . (A_{+},A_{-})\longmapsto\frac{A_{+}+A_{-}}{2}+x\frac{A_{+}-A_{-}}{2r(b')}. ( A + , A − ) ⟼ 2 A + + A − + x 2 r ( b ′ ) A + − A − . which proves the splitting.
(iii) Put
R 2 : = q f ξ + q f 4 ∫ S c 2 + 1 8 ∫ S Δ c − γ q f . \mathcal{R}_{2}:=q_{f}\xi+\frac{q_{f}}{4}\int_{S}c^{2}+\frac{1}{8}\int_{S}\Delta c-\frac{\gamma}{q_{f}}. R 2 := q f ξ + 4 q f ∫ S c 2 + 8 1 ∫ S Δ c − q f γ . Expanding the finite exponentials and using D S p = p 2 = 0 D_{S}p=p^{2}=0 D S p = p 2 = 0
R ( b ′ ) = 2 q f 1 + q f c + R 2 p , \mathcal{R}(b')=2q_{f}1+q_{f}c+\mathcal{R}_{2}p, R ( b ′ ) = 2 q f 1 + q f c + R 2 p , and
r ( b ′ ) ( 21 − c − 2 ξ p ) = 2 q f 1 + 1 4 Δ − R 2 p . r(b')(2\mathbf{1}-c-2\xi p)=2q_{f}1+\frac{1}{4}\Delta-\mathcal{R}_{2}p. r ( b ′ ) ( 2 1 − c − 2 ξ p ) = 2 q f 1 + 4 1 Δ − R 2 p . Applying Φ \Phi Φ (3) , we obtain
Eu ± = ( u ± 2 q f ) 1 + ( c 1 ( T S ) − 1 4 Δ ± 1 4 Δ ) − ( w ± R 2 ) p . \operatorname{Eu}_{\pm}=(u\pm2q_{f})1+\left(c_{1}(T_{S})-\frac{1}{4}\Delta\pm\frac{1}{4}\Delta\right)-(w\pm\mathcal{R}_{2})p. Eu ± = ( u ± 2 q f ) 1 + ( c 1 ( T S ) − 4 1 Δ ± 4 1 Δ ) − ( w ± R 2 ) p . Since
( t ± ) 0 = u ± 2 q f , ( t ± ) 2 = w ± R 2 , c 1 ( T S ) − 1 2 Δ = − 1 2 Δ S , (t_{\pm})_{0}=u\pm2q_{f},\qquad(t_{\pm})_{2}=w\pm\mathcal{R}_{2},\qquad c_{1}(T_{S})-\frac{1}{2}\Delta=-\frac{1}{2}\Delta_{S}, ( t ± ) 0 = u ± 2 q f , ( t ± ) 2 = w ± R 2 , c 1 ( T S ) − 2 1 Δ = − 2 1 Δ S , this proves (5) . □
Coefficients and divisor variables
Our next step is to compare the vertical product with the full quantum product. We first choose coordinates in which the origin of the negative factor is at zero. We then choose a coefficient ring in which this change of coordinates is well-defined and the vertical product is recovered by setting the contributions of nonvertical curves equal to zero.
At c = ξ = 0 c=\xi=0 c = ξ = 0 Proposition 3.3 gives
t − = ( u − 2 q f ) 1 + ( w + γ / q f ) p . t_{-}=(u-2q_{f})1+(w+\gamma/q_{f})p. t − = ( u − 2 q f ) 1 + ( w + γ / q f ) p . Thus the origin of this factor corresponds to u = 2 q f u=2q_{f} u = 2 q f w = − γ / q f w=-\gamma/q_{f} w = − γ / q f
u = 2 q f + u ′ , w = − γ q f + w ′ . (6) u=2q_{f}+u',\qquad w=-\frac{\gamma}{q_{f}}+w'. \tag*{(6)} u = 2 q f + u ′ , w = − q f γ + w ′ . (6) In particular, t − = 0 t_{-}=0 t − = 0 u ′ = w ′ = c = ξ = 0 u'=w'=c=\xi=0 u ′ = w ′ = c = ξ = 0
This substitution requires some care. A formal power series in w w w w = − γ / q f + w ′ w=-\gamma/q_f+w' w = − γ / q f + w ′ w ′ w' w ′ w w w
To keep track of projected curve classes, put
L : = π ∗ N 1 ( X , Z ) , P : = π ∗ NE ( X , Z ) ⊆ L . L:=\pi_*N_1(X,\mathbb{Z}), \qquad P:=\pi_*\operatorname{NE}(X,\mathbb{Z})\subseteq L. L := π ∗ N 1 ( X , Z ) , P := π ∗ NE ( X , Z ) ⊆ L . Thus L L L S S S P P P X X X
Since
ker ( π ∗ : N 1 ( X , Z ) → L ) = Z ℓ \ker(\pi_*:N_1(X,\mathbb{Z})\to L)=\mathbb{Z}\ell ker ( π ∗ : N 1 ( X , Z ) → L ) = Z ℓ and L L L
σ : L ⟶ N 1 ( X , Z ) , π ∗ ∘ σ = id L . \sigma:L\longrightarrow N_1(X,\mathbb{Z}), \qquad\pi_*\circ\sigma=\operatorname{id}_L. σ : L ⟶ N 1 ( X , Z ) , π ∗ ∘ σ = id L . For example, choose a basis of L L L N 1 ( X , Z ) N_1(X,\mathbb{Z}) N 1 ( X , Z ) B B B
B = σ ( β ) + m ℓ , β = π ∗ B , m ∈ Z . B=\sigma(\beta)+m\ell, \qquad\beta=\pi_*B,\quad m\in\mathbb{Z}. B = σ ( β ) + m ℓ , β = π ∗ B , m ∈ Z . We accordingly write
q B = q S β q f m . q^B=q_S^\beta q_f^m. q B = q S β q f m . The symbols q S β q_S^\beta q S β σ ( β ) \sigma(\beta) σ ( β ) m m m B B B A S A_S A S
l ( β ) : = ∫ β A S . l(\beta):=\int_\beta A_S. l ( β ) := ∫ β A S . Every nonzero class in P P P l l l P P P l l l
We now justify the required polynomial dependence on w w w w w w p p p
Fix β ∈ P ∖ { 0 } \beta\in P\setminus\{0\} β ∈ P ∖ { 0 } β \beta β l ( β ) l(\beta) l ( β ) S S S β ≠ 0 \beta\ne0 β = 0 π \pi π
M β : = dim Chow 1 , l ( β ) ( S ) , M_\beta:=\dim\operatorname{Chow}_{1,l(\beta)}(S), M β := dim Chow 1 , l ( β ) ( S ) , where the Chow variety parametrizes effective one-cycles of this degree for the embedding defined by A S A_S A S
Consider the incidence variety parametrizing such a cycle together with k k k M β + k M_\beta+k M β + k dim S k = 2 k \dim S^k=2k dim S k = 2 k k > M β k>M_\beta k > M β S k S^k S k k k k S S S β \beta β M β M_\beta M β p p p
Thus, at fixed β ≠ 0 \beta\ne0 β = 0 w w w β = 0 \beta=0 β = 0 u u u u , w u,w u , w κ t ′ \kappa_{t'} κ t ′
There is likewise no infinite summation over the fiber degree once the projected class and parameter monomial are fixed. Indeed, homogeneity determines the fiber degree. If a homogeneous coefficient has degree d d d
q S β q f m u a w b ( c 1 ) I 1 ⋯ ( c p S ) I p S ξ j q_S^\beta q_f^m u^a w^b(c^1)^{I_1}\cdots(c^{p^S})^{I_{p^S}}\xi^j q S β q f m u a w b ( c 1 ) I 1 ⋯ ( c p S ) I p S ξ j can occur only if
d = 2 ∫ σ ( β ) ( c 1 ( T X ) + 2 m + 2 a − 2 b − 2 ∣ I ∣ − 4 j ) , ∣ I ∣ : = ∑ i I i . d = 2\int_{\sigma(\beta)} \left(c_1(T_X) + 2m + 2a - 2b - 2\lvert I\rvert- 4j\right), \qquad\lvert I\rvert:= \sum_i I_i. d = 2 ∫ σ ( β ) ( c 1 ( T X ) + 2 m + 2 a − 2 b − 2 ∣ I ∣ − 4 j ) , ∣ I ∣ := i ∑ I i . Thus, for fixed β \beta β m m m
Recall that Nov X ∘ \operatorname{Nov}^{\circ}_X Nov X ∘ Q \mathbb{Q} Q f ∈ Nov X ∘ f \in\operatorname{Nov}^{\circ}_X f ∈ Nov X ∘
f = ∑ β ∈ P q S β f β , f β ∈ Q [ q f , q f − 1 ] [ [ c 1 , … , c ρ S , ξ ] ] [ u , w ] . f = \sum_{\beta\in P}q_S^\beta f_\beta,\qquad f_\beta\in\mathbb{Q}[q_f,q_f^{-1}][[c^1,\ldots,c^{\rho_S},\xi]][u,w]. f = β ∈ P ∑ q S β f β , f β ∈ Q [ q f , q f − 1 ] [[ c 1 , … , c ρ S , ξ ]] [ u , w ] . The brackets [ u , w ] [u,w] [ u , w ] f β f_\beta f β u , w u,w u , w c 1 , … , c ρ S , ξ c^1,\ldots,c^{\rho_S},\xi c 1 , … , c ρ S , ξ β ∈ P \beta\in P β ∈ P
We can now specify the new coefficient ring. Treat q f q_f q f
k f : = Q ( q f ) , R : = k f [ [ P ] ] l = { ∑ β ∈ P a β q S β : a β ∈ k f } . k_f := \mathbb{Q}(q_f),\qquad R := k_f[[P]]_l = \left\{\sum_{\beta\in P}a_\beta q_S^\beta: a_\beta\in k_f\right\}. k f := Q ( q f ) , R := k f [[ P ] ] l = ⎩ ⎨ ⎧ β ∈ P ∑ a β q S β : a β ∈ k f ⎭ ⎬ ⎫ . Multiplication is given by q S β q S β ′ = q S β + β ′ q_S^\beta q_S^{\beta'}=q_S^{\beta+\beta'} q S β q S β ′ = q S β + β ′
F r R : = { ∑ β a β q S β : a β = 0 whenever l ( β ) < r } , r ≥ 0. F^rR := \left\{\sum_\beta a_\beta q_S^\beta: a_\beta=0\ \text{whenever}\ l(\beta)<r\right\},\qquad r\geq0. F r R := ⎩ ⎨ ⎧ β ∑ a β q S β : a β = 0 whenever l ( β ) < r ⎭ ⎬ ⎫ , r ≥ 0. Thus, F r R F^rR F r R r r r R R R
R / F 1 R = k f , F r R F s R ⊆ F r + s R . R/F^1R=k_f,\qquad F^rR\,F^sR\subseteq F^{r+s}R. R / F 1 R = k f , F r R F s R ⊆ F r + s R . In particular, reduction modulo F 1 R F^1R F 1 R
Writing c = ( c 1 , … , c ρ S ) \mathbf{c}=(c^1,\ldots,c^{\rho_S}) c = ( c 1 , … , c ρ S )
Nov X ∘ ⟶ R [ [ u ′ , w ′ , c , ξ ] ] \operatorname{Nov}^{\circ}_X\longrightarrow R[[u',w',\mathbf{c},\xi]] Nov X ∘ ⟶ R [[ u ′ , w ′ , c , ξ ]] by applying (6) to each polynomial f β f_\beta f β β \beta β
u ′ = u − 2 q f , w ′ = w + γ / q f . u'=u-2q_f,\qquad w'=w+\gamma/q_f. u ′ = u − 2 q f , w ′ = w + γ / q f . We have therefore justified the change of origin on the algebra containing all coefficients needed for the quantum product and κ t ′ \kappa_{t'} κ t ′
We next restore the divisor parameters. This allows us to represent all cohomology classes by coordinate vector fields, including the divisor classes that were absorbed into the Novikov variables in the reduced description.
Introduce formal coordinates v 1 , … , v ρ S , z v^1,\ldots,v^{\rho_S},z v 1 , … , v ρ S , z Δ 1 , … , Δ ρ S , x \Delta_1,\ldots,\Delta_{\rho_S},x Δ 1 , … , Δ ρ S , x
v : = ∑ i v i Δ i , v : = ( v 1 , … , v ρ S ) , O ^ : = R [ [ u ′ , w ′ , v , z , c , ξ ] ] . v:=\sum_i v^i\Delta_i,\qquad\mathbf{v}:=(v^1,\ldots,v^{\rho_S}),\qquad\widehat{\mathcal{O}}:=R[[u',w',\mathbf{v},z,\mathbf{c},\xi]]. v := i ∑ v i Δ i , v := ( v 1 , … , v ρ S ) , O := R [[ u ′ , w ′ , v , z , c , ξ ]] . The full formal parameter is
a = ( 2 q f + u ′ ) 1 + v + ( − γ q f + w ′ ) p , b = z 1 + c + ξ p . (7) a=(2q_f+u')\mathbf{1}+v+\left(-\frac{\gamma}{q_f}+w'\right)p,
\qquad
b=z\mathbf{1}+\mathbf{c}+\xi p. \tag*{(7)} a = ( 2 q f + u ′ ) 1 + v + ( − q f γ + w ′ ) p , b = z 1 + c + ξ p . (7) The divisor axiom describes exactly how the new parameters enter the product. In curve class B B B v + z x v+zx v + z x
∫ B ( v + z x ) = ∫ π ∗ B v + z ∫ B x . \int_B(v+zx)=\int_{\pi_*B}v+z\int_Bx. ∫ B ( v + z x ) = ∫ π ∗ B v + z ∫ B x . Summing over any number of these insertions gives the exponential of this expression. Consequently, the coefficients of the quantum product at t t t
q B ⟼ q s β q f m exp ( ∫ β v + z ∫ B x ) , B = σ ( β ) + m ℓ , q^B\longmapsto q_s^\beta q_f^m\exp\left(\int_\beta v+z\int_Bx\right),\qquad B=\sigma(\beta)+m\ell, q B ⟼ q s β q f m exp ( ∫ β v + z ∫ B x ) , B = σ ( β ) + m ℓ , u ⟼ 2 q f + u ′ , u\longmapsto2q_f+u', u ⟼ 2 q f + u ′ , w ⟼ − γ q f + w ′ , (8) w\longmapsto-\frac{\gamma}{q_f}+w', \tag*{(8)} w ⟼ − q f γ + w ′ , (8) with c i , ξ c^i,\xi c i , ξ
The preceding finiteness arguments make this substitution well-defined on Nov X ∘ \operatorname{Nov}^{\circ}_X Nov X ∘ v = z = 0 v=z=0 v = z = 0 (8) is injective as well.
Since divisor parameters have coefficient zero in the linear part of the Euler vector field, the same substitution sends the matrix of κ t ′ \kappa_{t'} κ t ′ t t t q f q_f q f R R R z z z u = 2 q f u=2q_f u = 2 q f w = − γ / q f w=-\gamma/q_f w = − γ / q f
Finally, we specify the topology used for subsequent formal coordinate changes. For a finite tuple of variables y \mathbf y y F r R [ [ y ] ] F^rR[[\mathbf y]] F r R [[ y ]] F r R F^rR F r R R [ [ y ] ] R[[\mathbf y]] R [[ y ]]
F r R [ [ y ] ] + ( y ) r , r ≥ 1. F^rR[[\mathbf y]]+(\mathbf y)^r,\qquad r\geq1. F r R [[ y ]] + ( y ) r , r ≥ 1. Thus terms tend to zero when either their projected curve degree or their total parameter degree tends to infinity. In particular, a translation by a coefficient in F 1 R F^1R F 1 R F F F
Extending the vertical decomposition
We now work over the coefficient ring O ^ \widehat{\mathcal O} O F 1 F^1 F 1
Write ⋆ \star ⋆ (7) . The restored divisor coordinates allow us to regard this product as a multiplication of coordinate vector fields. More precisely, let
T : = Der R c o n t ( O ^ ) ≃ H e v e n ( X , Q ) ⊗ Q O ^ , \mathcal T:=\operatorname{Der}^{\mathrm{cont}}_R(\widehat{\mathcal O})\simeq H^{\mathrm{even}}(X,\mathbb Q)\otimes_{\mathbb Q}\widehat{\mathcal O}, T := Der R cont ( O ) ≃ H even ( X , Q ) ⊗ Q O , where an R R R R R R
∂ u ′ ⟷ 1 , ∂ v i ⟷ Δ i , ∂ z ⟷ x , \partial_{u'}\longleftrightarrow1,\qquad
\partial_{v_i}\longleftrightarrow\Delta_i,\qquad
\partial_z\longleftrightarrow x, ∂ u ′ ⟷ 1 , ∂ v i ⟷ Δ i , ∂ z ⟷ x , ∂ w ′ ⟷ p , ∂ c i ⟷ x Δ i , ∂ ξ ⟷ x p . \partial_{w'}\longleftrightarrow p,\qquad
\partial_{c_i}\longleftrightarrow x\Delta_i,\qquad
\partial_\xi\longleftrightarrow xp. ∂ w ′ ⟷ p , ∂ c i ⟷ x Δ i , ∂ ξ ⟷ x p . In particular, the unit of ⋆ \star ⋆ ∂ u ′ \partial_{u'} ∂ u ′ 1 1 1
We first describe the vertical decomposition in these coordinates. The formulas of Proposition 3.3 extend to b = z 1 + c + ξ p b=z\mathbf1+c+\xi p b = z 1 + c + ξ p
r ( b ) : = q f e b / 2 + 1 8 Δ + η 2 q f e − b / 2 , r(b):=q_f e^{b/2}+\frac{1}{8}\Delta+\frac{\eta}{2q_f}e^{-b/2}, r ( b ) := q f e b /2 + 8 1 Δ + 2 q f η e − b /2 , R ( b ) : = 2 q f e b / 2 + 1 8 Δ b − η q f e − b / 2 , \mathcal R(b):=2q_f e^{b/2}+\frac{1}{8}\Delta b-\frac{\eta}{q_f}e^{-b/2}, R ( b ) := 2 q f e b /2 + 8 1 Δ b − q f η e − b /2 , t ± : = a ± R ( b ) . t_\pm:=a\pm\mathcal R(b). t ± := a ± R ( b ) . Here η = γ p \eta=\gamma p η = γ p e ± b / 2 = e ± z / 2 e ± ( c + ξ p ) / 2 e^{\pm b/2}=e^{\pm z/2}e^{\pm(c+\xi p)/2} e ± b /2 = e ± z /2 e ± ( c + ξ p ) /2 r ( b ) r(b) r ( b ) q f e z / 2 q_f e^{z/2} q f e z /2 r ( b ) r(b) r ( b )
ϵ ± ( 0 ) = 1 2 ( 1 ± r ( b ) − 1 x ) . \epsilon_{\pm}^{(0)}=\frac{1}{2}\left(1\pm r(b)^{-1}x\right). ϵ ± ( 0 ) = 2 1 ( 1 ± r ( b ) − 1 x ) . They satisfy
ϵ ± ( 0 ) ⋆ v e r t ϵ ± ( 0 ) = ϵ ± ( 0 ) , ϵ + ( 0 ) ⋆ v e r t ϵ − ( 0 ) = 0 , ϵ + ( 0 ) + ϵ − ( 0 ) = 1. \epsilon_{\pm}^{(0)}\star_{\mathrm{vert}}\epsilon_{\pm}^{(0)}=\epsilon_{\pm}^{(0)},\qquad\epsilon_{+}^{(0)}\star_{\mathrm{vert}}\epsilon_{-}^{(0)}=0,\qquad\epsilon_{+}^{(0)}+\epsilon_{-}^{(0)}=1. ϵ ± ( 0 ) ⋆ vert ϵ ± ( 0 ) = ϵ ± ( 0 ) , ϵ + ( 0 ) ⋆ vert ϵ − ( 0 ) = 0 , ϵ + ( 0 ) + ϵ − ( 0 ) = 1. Multiplication by these idempotents projects onto the two vertical summands.
The same decomposition can be expressed in terms of a change of coordinates. Indeed,
d R b ( δ b ) = r ( b ) δ b , dR_b(\delta b)=r(b)\delta b, d R b ( δ b ) = r ( b ) δ b , so the differential of ( a , b ) ↦ ( t + , t − ) (a,b)\mapsto(t_+,t_-) ( a , b ) ↦ ( t + , t − )
( δ a , δ b ) ⟼ ( δ a + r ( b ) δ b , δ a − r ( b ) δ b ) . (\delta a,\delta b)\longmapsto(\delta a+r(b)\delta b,\delta a-r(b)\delta b). ( δ a , δ b ) ⟼ ( δ a + r ( b ) δ b , δ a − r ( b ) δ b ) . This is precisely the vertical algebra isomorphism of Proposition 3.3. In these coordinates, each summand has the constant cup product of H S H_S H S
t − = 0 , t + = 4 q f 1 − 2 γ q f p . t_-=0,\qquad t_+=4q_f\mathbf{1}-\frac{2\gamma}{q_f}p. t − = 0 , t + = 4 q f 1 − q f 2 γ p . Hence t − t_- t − t + − ( 4 q f 1 − 2 γ p / q f ) t_+-(4q_f\mathbf{1}-2\gamma p/q_f) t + − ( 4 q f 1 − 2 γ p / q f )
We now restore the contributions of nonvertical curves. The idempotents ϵ ± ( 0 ) \epsilon_{\pm}^{(0)} ϵ ± ( 0 ) ϵ ± \epsilon_{\pm} ϵ ± ⋆ \star ⋆
ϵ ⋆ ϵ − ϵ = 0 \epsilon\star\epsilon-\epsilon=0 ϵ ⋆ ϵ − ϵ = 0 successively in the filtration F r F^r F r h h h
( 2 ϵ − 1 ) ⋆ h = − ( ϵ ⋆ ϵ − ϵ ) . (2\epsilon-1)\star h=-(\epsilon\star\epsilon-\epsilon). ( 2 ϵ − 1 ) ⋆ h = − ( ϵ ⋆ ϵ − ϵ ) . Its coefficient is invertible: modulo F 1 F^1 F 1 ( 2 ϵ ± ( 0 ) − 1 ) 2 = 1 (2\epsilon_{\pm}^{(0)}-1)^2=1 ( 2 ϵ ± ( 0 ) − 1 ) 2 = 1 ϵ − = 1 − ϵ + \epsilon_-=1-\epsilon_+ ϵ − = 1 − ϵ +
ϵ + ⋆ ϵ − = 0 , T = T + ⊕ T − , T ± : = ϵ ± ⋆ T . \epsilon_+\star\epsilon_-=0,\qquad\mathcal{T}=\mathcal{T}_+\oplus\mathcal{T}_-,\qquad\mathcal{T}_{\pm}:=\epsilon_{\pm}\star\mathcal{T}. ϵ + ⋆ ϵ − = 0 , T = T + ⊕ T − , T ± := ϵ ± ⋆ T . Both summands have rank ρ S + 2 \rho_S+2 ρ S + 2
This gives a decomposition of the algebra of vector fields. To achieve this using two groups of coordinates, we use the compatibility between the quantum product and the Lie bracket. For vector fields Y , V , W Y,V,W Y , V , W
P Y ( V , W ) : = [ Y , V ⋆ W ] − [ Y , V ] ⋆ W − V ⋆ [ Y , W ] . \mathcal{P}_Y(V,W):=[Y,V\star W]-[Y,V]\star W-V\star[Y,W]. P Y ( V , W ) := [ Y , V ⋆ W ] − [ Y , V ] ⋆ W − V ⋆ [ Y , W ] . Thus P Y = Lie Y ( ⋆ ) \mathcal{P}_Y=\operatorname{Lie}_Y(\star) P Y = Lie Y ( ⋆ ) Y Y Y
P Y ⋆ Z = Y ⋆ P Z + Z ⋆ P Y ; (9) \mathcal{P}_{Y\star Z}=Y\star\mathcal{P}_Z+Z\star\mathcal{P}_Y; \tag*{(9)} P Y ⋆ Z = Y ⋆ P Z + Z ⋆ P Y ; (9) see [10 Theorem 2(a)] . A formal power-series space over R R R formal F F F R R R .
For an idempotent ϵ \epsilon ϵ P ϵ = 2 ϵ ⋆ P ϵ \mathcal{P}_{\epsilon}=2\epsilon\star\mathcal{P}_{\epsilon} P ϵ = 2 ϵ ⋆ P ϵ 1 − 2 ϵ 1-2\epsilon 1 − 2 ϵ P ϵ = 0 \mathcal{P}_{\epsilon}=0 P ϵ = 0 Y , Z ∈ T + Y,Z\in\mathcal{T}_+ Y , Z ∈ T + ϵ − ⋆ Y = 0 \epsilon_-\star Y=0 ϵ − ⋆ Y = 0 ( ϵ − , Z ) (\epsilon_-,Z) ( ϵ − , Z )
0 = ϵ − ⋆ P Y ( ϵ − , Z ) = − ϵ − ⋆ [ Y , Z ] . 0=\epsilon_-\star\mathcal{P}_Y(\epsilon_-,Z)=-\epsilon_-\star[Y,Z]. 0 = ϵ − ⋆ P Y ( ϵ − , Z ) = − ϵ − ⋆ [ Y , Z ] . Hence [ Y , Z ] ∈ T + [Y,Z]\in\mathcal{T}_+ [ Y , Z ] ∈ T + T − \mathcal{T}_- T −
Moreover, the product within either group is independent of the other group’s coordinates. For example, let A A A B B B T + \mathcal{T}_{+} T + Z Z Z T − \mathcal{T}_{-} T − Z = ϵ − ⋆ Z Z=\epsilon_{-}\star Z Z = ϵ − ⋆ Z P ϵ − = 0 \mathcal{P}_{\epsilon_{-}}=0 P ϵ − = 0
P Z ( A , B ) = ϵ − ⋆ P Z ( A , B ) . \mathcal{P}_{Z}(A,B)=\epsilon_{-}\star\mathcal{P}_{Z}(A,B). P Z ( A , B ) = ϵ − ⋆ P Z ( A , B ) . The coordinate fields commute, so the left-hand side is [ Z , A ⋆ B ] [Z,A\star B] [ Z , A ⋆ B ] T + \mathcal{T}_{+} T + T − \mathcal{T}_{-} T −
This is the decomposition argument in the proof of [9 Theorem 4.2] , applied formally over R R R F r F^{r} F r F F F
We next construct a derivation suited to the negative factor. The reason for modifying the Euler vector field is already visible in the vertical calculation: its negative component contains the constant divisor term
C : = − 1 2 Δ S = c 1 ( T S ) − 1 2 Δ . C:=-\frac{1}{2}\Delta_{S}=c_{1}(T_{S})-\frac{1}{2}\Delta. C := − 2 1 Δ S = c 1 ( T S ) − 2 1 Δ. We shall replace differentiation in this divisor direction by an equivalent action on the curve coefficients. Nefness will then control that action.
For D = ∑ i D i Δ i ∈ D S D=\sum_{i}D^{i}\Delta_{i}\in D_{S} D = ∑ i D i Δ i ∈ D S
∂ v , D : = ∑ i D i ∂ v i , δ D ( q S β ) : = ( D ⋅ β ) q S β . \partial_{v,D}:=\sum_{i}D^{i}\partial_{v_{i}},\qquad\delta_{D}(q_{S}^{\beta}):=(D\cdot\beta)q_{S}^{\beta}. ∂ v , D := i ∑ D i ∂ v i , δ D ( q S β ) := ( D ⋅ β ) q S β . Here δ D \delta_{D} δ D k F k_{F} k F R R R O ^ \widehat{\mathcal{O}} O (8) shows that differentiating a product coefficient along ∂ v , D \partial_{v,D} ∂ v , D δ D \delta_{D} δ D
Lie ∂ v , D − δ D ( ⋆ ) = 0. \operatorname{Lie}_{\partial_{v,D}-\delta_{D}}(\star)=0. Lie ∂ v , D − δ D ( ⋆ ) = 0. Define
S C : = ∂ v , C − δ C , G : = E u t − S C . S_{C}:=\partial_{v,C}-\delta_{C},\qquad\mathcal{G}:=E_{\mathrm{ut}}-S_{C}. S C := ∂ v , C − δ C , G := E ut − S C . In our coordinates, the Euler vector field is
E u t = u ∂ u ′ + ∂ v , c 1 ( T S ) − Δ / 4 + 2 ∂ z − w ∂ w ′ − ∑ i c i ∂ z i − 2 ξ ∂ ξ , E_{\mathrm{ut}}=u\partial_{u'}+\partial_{v,c_{1}(T_{S})-\Delta/4}+2\partial_{z}-w\partial_{w'}-\sum_{i}c^{i}\partial_{z_{i}}-2\xi\partial_{\xi}, E ut = u ∂ u ′ + ∂ v , c 1 ( T S ) − Δ/4 + 2 ∂ z − w ∂ w ′ − i ∑ c i ∂ z i − 2 ξ ∂ ξ , where u = 2 q F + u ′ u=2q_{F}+u' u = 2 q F + u ′ w = − γ / q F + w ′ w=-\gamma/q_{F}+w' w = − γ / q F + w ′ R R R
G ∣ R = δ C , Lie G ( ⋆ ) = ⋆ . (10) \mathcal{G}\big|_{R}=\delta_{C},\qquad\operatorname{Lie}_{\mathcal{G}}(\star)=\star. \tag*{(10)} G R = δ C , Lie G ( ⋆ ) = ⋆ . (10) Although G \mathcal{G} G R R R R R R R R R
The derivation G \mathcal{G} G (10) to an idempotent ϵ \epsilon ϵ
[ G , ϵ ] − 2 ϵ ⋆ [ G , ϵ ] = ϵ , [\mathcal{G},\epsilon]-2\epsilon\star[\mathcal{G},\epsilon]=\epsilon, [ G , ϵ ] − 2 ϵ ⋆ [ G , ϵ ] = ϵ , and hence [ G , ϵ ] = − ϵ [\mathcal{G},\epsilon]=-\epsilon [ G , ϵ ] = − ϵ V ∈ T V\in\mathcal{T} V ∈ T
[ G , ϵ ⋆ V ] = ϵ ⋆ [ G , V ] . [\mathcal{G},\epsilon\star V]=\epsilon\star[\mathcal{G},V]. [ G , ϵ ⋆ V ] = ϵ ⋆ [ G , V ] . In product coordinates, this means that the coefficients of G \mathcal{G} G R R R G \mathcal{G} G R R R δ C \delta_{C} δ C
To compute the induced derivation on the negative factor modulo F 1 F^{1} F 1
t − = U 1 + ∑ i V i Δ i + W p , C = ∑ i c i Δ i . t_{-}=U1+\sum_{i}V_{i}\Delta_{i}+Wp,\qquad C=\sum_{i}c^{i}\Delta_{i}. t − = U 1 + i ∑ V i Δ i + W p , C = i ∑ c i Δ i . The calculation of (5) remains valid with b = z 1 + c + ξ p b=z_1+c+\xi_p b = z 1 + c + ξ p
U ∂ U + ∑ i C i ∂ V i − W ∂ W . U\partial_U+\sum_i C^i\partial_{V_i}-W\partial_W. U ∂ U + i ∑ C i ∂ V i − W ∂ W . Indeed, replacing q f q_f q f q f e z / 2 q_f e^{z/2} q f e z /2 z Δ / 8 z\Delta/8 z Δ/8 R ( b ) R(b) R ( b ) t − t_- t − ∂ v , C \partial_{v,C} ∂ v , C t − t_- t − C C C ∑ i C i ∂ V i \sum_i C^i\partial_{V_i} ∑ i C i ∂ V i δ C \delta_C δ C U , V i , W U,V_i,W U , V i , W
O − : = R [ [ U , V 1 , … , V ρ S , W ] ] . \mathcal{O}_-:=R[[U,V_1,\ldots,V_{\rho_S},W]]. O − := R [[ U , V 1 , … , V ρ S , W ]] . Write ∘ \circ ∘ G − \mathcal{G}_- G − O − \mathcal{O}_- O − T − \mathcal{T}_- T −
G − ≡ G 0 : = δ C + U ∂ U − W ∂ W ( m o d F 1 ) , Lie G − ( ∘ ) = ∘ . (11) \mathcal{G}_-\equiv\mathcal{G}_0:=\delta_C+U\partial_U-W\partial_W\pmod{F^1},\qquad\operatorname{Lie}_{\mathcal{G}_-}(\circ)=\circ. \tag*{(11)} G − ≡ G 0 := δ C + U ∂ U − W ∂ W ( mod F 1 ) , Lie G − ( ∘ ) = ∘ . (11) The congruence concerns coefficients in the coordinate directions; both derivations restrict to δ C \delta_C δ C R R R F 1 F^1 F 1 H S H_S H S
∂ V i ∘ ∂ V j = g i j ∂ W , ∂ V i ∘ ∂ W = ∂ W ∘ ∂ W = 0 , \partial_{V_i}\circ\partial_{V_j}=g_{ij}\partial_W,\qquad\partial_{V_i}\circ\partial_W=\partial_W\circ\partial_W=0, ∂ V i ∘ ∂ V j = g ij ∂ W , ∂ V i ∘ ∂ W = ∂ W ∘ ∂ W = 0 , and its unit reduces to ∂ U \partial_U ∂ U
Nefness and non-splitting
The preceding construction gives two summands of the full quantum algebra. We must still prove that the negative summand is local after passing to the algebraically closed fraction field.
Remark 3.4. For the discussion that follows, it is important to note that locality of the negative factor in vertical quantum cohomology does not immediately imply locality after the contributions of the remaining curve classes are restored. A simple toy model is Q [ [ τ ] ] [ ζ ] / ( ζ 2 − τ ) \mathbb{Q[[\tau]][\zeta]/(\zeta^2-\tau)} Q [[ τ ]] [ ζ ] / ( ζ 2 − τ ) τ = 0 \tau=0 τ = 0 Q ( ( τ ) ) \mathbb{Q((\tau))} Q (( τ ))
We will see that nefness rules out the above kind of splitting in our situation. Namely, δ C \delta_C δ C
δ C ( q S β ) = λ β q S β , λ β : = C ⋅ β = − 1 2 Δ S ⋅ β ≤ 0. \delta_C(q_S^\beta)=\lambda_\beta q_S^\beta,\qquad\lambda_\beta:=C\cdot\beta=-\frac{1}{2}\Delta_S\cdot\beta\leq0. δ C ( q S β ) = λ β q S β , λ β := C ⋅ β = − 2 1 Δ S ⋅ β ≤ 0. The next lemma combines this inequality with (11) to constrain the possible nonvertical corrections to the product. It shows that, after a formal change of coordinates, the products among the nonunit directions retain the same form as in H S H_S H S
Lemma 3.5. Let R = k f [ [ P ] ] l R=k_f[[P]]_l R = k f [[ P ] ] l δ \delta δ k f k_f k f
δ ( q S β ) = λ β q S β \delta(q_S^\beta)=\lambda_\beta q_S^\beta δ ( q S β ) = λ β q S β where λ : P → Q \lambda:P\to\mathbb{Q} λ : P → Q λ β ≤ 0 \lambda_\beta\leq0 λ β ≤ 0
O : = R [ [ U , V 1 , … , V s , W ] ] , s ≥ 1 , \mathcal{O}:=R[[U,V_1,\ldots,V_s,W]],\qquad s\geq1, O := R [[ U , V 1 , … , V s , W ]] , s ≥ 1 , carry a formal F F F ∘ \circ ∘ e e e F 1 F^1 F 1
e = ∂ U , ∂ V i ∘ ∂ V j = g i j ∂ W , ∂ V i ∘ ∂ W = ∂ W ∘ ∂ W = 0 , e=\partial_U,\qquad\partial_{V_i}\circ\partial_{V_j}=g_{ij}\partial_W,\qquad\partial_{V_i}\circ\partial_W=\partial_W\circ\partial_W=0, e = ∂ U , ∂ V i ∘ ∂ V j = g ij ∂ W , ∂ V i ∘ ∂ W = ∂ W ∘ ∂ W = 0 , where g = ( g i j ) g=(g_{ij}) g = ( g ij ) G \mathcal{G} G
G ∣ R = δ , Lie G ( ∘ ) = ∘ , G ≡ δ + U ∂ U − W ∂ W ( m o d F 1 ) . \mathcal{G}|_R=\delta,\qquad\operatorname{Lie}_{\mathcal{G}}(\circ)=\circ,\qquad\mathcal{G}\equiv\delta+U\partial_U-W\partial_W\pmod{F^1}. G ∣ R = δ , Lie G ( ∘ ) = ∘ , G ≡ δ + U ∂ U − W ∂ W ( mod F 1 ) . Then a formal coordinate change over R R R F 1 F^1 F 1
e = ∂ U , ∂ V i ∘ ∂ V j = G i j ∂ W , ∂ V i ∘ ∂ W = 0 , ∂ W ∘ ∂ W = 0. (12) \begin{aligned}
e &= \partial_U, & \partial_{V_i} \circ\partial_{V_j} &= G_{ij}\partial_W,\\
\partial_{V_i} \circ\partial_W &= 0, & \partial_W \circ\partial_W &= 0. \tag*{(12)}
\end{aligned} e ∂ V i ∘ ∂ W = ∂ U , = 0 , ∂ V i ∘ ∂ V j ∂ W ∘ ∂ W = G ij ∂ W , = 0. (12) The matrix G = ( G i j ) G=(G_{ij}) G = ( G ij ) G ≡ g ( m o d F 1 ) G \equiv g \pmod{F^1} G ≡ g ( mod F 1 ) U , W U,W U , W β \beta β λ β = 0 \lambda_\beta=0 λ β = 0
After extending coefficients to Ω : = Frac ( O ) \Omega:= \operatorname{Frac}(\mathcal{O}) Ω := Frac ( O ) s + 2 s+2 s + 2 s + 1 s+1 s + 1
Proof. We first choose coordinates in which the unit is constant. We then simplify G \mathcal{G} G Lie G ( ∘ ) = ∘ \operatorname{Lie}_{\mathcal{G}}(\circ)=\circ Lie G ( ∘ ) = ∘
Since e ≡ ∂ U ( m o d F 1 ) e \equiv\partial_U \pmod{F^1} e ≡ ∂ U ( mod F 1 ) e e e F 1 F^1 F 1 e = ∂ U e=\partial_U e = ∂ U e e e Lie e ( ∘ ) = 2 Lie e ( ∘ ) \operatorname{Lie}_e(\circ)=2\operatorname{Lie}_e(\circ) Lie e ( ∘ ) = 2 Lie e ( ∘ )
Lie ∂ U ( ∘ ) = 0. \operatorname{Lie}_{\partial_U}(\circ)=0. Lie ∂ U ( ∘ ) = 0. Thus every product coefficient is independent of U U U Lie G ( ∘ ) = ∘ \operatorname{Lie}_{\mathcal{G}}(\circ)=\circ Lie G ( ∘ ) = ∘ [ G , e ] = − e [\mathcal{G},e]=-e [ G , e ] = − e
G = G 0 + N , G 0 : = δ + U ∂ U − W ∂ W , \mathcal{G}=\mathcal{G}_0+N,\qquad\mathcal{G}_0:=\delta+U\partial_U-W\partial_W, G = G 0 + N , G 0 := δ + U ∂ U − W ∂ W , where N N N R R R F 1 F^1 F 1 U U U
To simplify N N N
ω U = 1 , ω V i = 0 , ω W = − 1. \omega_U=1,\qquad\omega_{V_i}=0,\qquad\omega_W=-1. ω U = 1 , ω V i = 0 , ω W = − 1. These are weights for the present calculation, not the cohomological degrees used earlier. With V ν : = ∏ i V i ν i V^\nu:=\prod_i V_i^{\nu_i} V ν := ∏ i V i ν i y j ∈ { U , V 1 , … , V s , W } y_j\in\{U,V_1,\ldots,V_s,W\} y j ∈ { U , V 1 , … , V s , W }
[ G 0 , q S β V ν W k ∂ y j ] = ( λ β − k − ω j ) q S β V ν W k ∂ y j . (13) [\mathcal{G}_0,q_S^\beta V^\nu W^k\partial_{y_j}]
=(\lambda_\beta-k-\omega_j)q_S^\beta V^\nu W^k\partial_{y_j}. \tag*{(13)} [ G 0 , q S β V ν W k ∂ y j ] = ( λ β − k − ω j ) q S β V ν W k ∂ y j . (13) The three contributions are the action of δ \delta δ q S β q_S^\beta q S β − W ∂ W -W\partial_W − W ∂ W W k W^k W k ∂ y j \partial_{y_j} ∂ y j
Write μ : = λ β − k − ω j \mu:=\lambda_\beta-k-\omega_j μ := λ β − k − ω j μ ≠ 0 \mu\ne0 μ = 0 a q S β V ν W k ∂ y j a q_S^\beta V^\nu W^k\partial_{y_j} a q S β V ν W k ∂ y j F F F
H : = a μ q S β V ν W k ∂ y j . H:=\frac{a}{\mu}q_S^\beta V^\nu W^k\partial_{y_j}. H := μ a q S β V ν W k ∂ y j . Indeed, [ H , G 0 ] = − a q S β V ν W k ∂ y j [H,\mathcal{G}_0]=-a q_S^\beta V^\nu W^k\partial_{y_j} [ H , G 0 ] = − a q S β V ν W k ∂ y j μ = 0 \mu=0 μ = 0 G 0 \mathcal{G}_0 G 0
At each order r ≥ 1 r\ge1 r ≥ 1 N N N F r + 1 F^{r+1} F r + 1 R R R H r ∈ F r H_r\in F^r H r ∈ F r exp ( H r ) \exp(H_r) exp ( H r ) G \mathcal{G} G exp ( ad H r ) G \exp(\operatorname{ad}H_r)\mathcal{G} exp ( ad H r ) G F r + 1 F^{r+1} F r + 1 N N N F r + 1 F^{r+1} F r + 1
Each change fixes R R R ∂ U \partial_U ∂ U H r H_r H r
G = G 0 + N , N ∈ F 1 , [ G 0 , N ] = 0. (14) \mathcal{G}=\mathcal{G}_0+N,\qquad N\in F^1,\qquad[\mathcal{G}_0,N]=0. \tag*{(14)} G = G 0 + N , N ∈ F 1 , [ G 0 , N ] = 0. (14) The correction N N N
Let M M M ( A , B ) ↦ A ∘ B (A,B)\mapsto A\circ B ( A , B ) ↦ A ∘ B
( Lie G 0 − 1 ) M = − Lie N M . (15) (\operatorname{Lie}_{G_0}-1)M=-\operatorname{Lie}_N M. \tag*{(15)} ( Lie G 0 − 1 ) M = − Lie N M . (15) Consider a term q S β V v W k q_S^\beta V^vW^k q S β V v W k ∂ y a \partial_{y_a} ∂ y a ∂ y i ∘ ∂ y j \partial_{y_i}\circ\partial_{y_j} ∂ y i ∘ ∂ y j Lie G 0 M \operatorname{Lie}_{G_0}M Lie G 0 M
λ β − k + ω i + ω j − ω a . \lambda_\beta-k+\omega_i+\omega_j-\omega_a. λ β − k + ω i + ω j − ω a . We call this number its tensor weight. The leading product, obtained modulo F 1 F^1 F 1
Because [ G 0 , N ] = 0 [G_0,N]=0 [ G 0 , N ] = 0 Lie N \operatorname{Lie}_N Lie N F F F N N N F 1 F^1 F 1 M M M M μ M_\mu M μ μ \mu μ μ ≠ 1 \mu\ne1 μ = 1 (15) gives
( μ − 1 ) M μ = − Lie N M μ . (\mu-1)M_\mu=-\operatorname{Lie}_N M_\mu. ( μ − 1 ) M μ = − Lie N M μ . Initially M μ ∈ F 1 M_\mu\in F^1 M μ ∈ F 1 M μ ∈ F r M_\mu\in F^r M μ ∈ F r F r + 1 F^{r+1} F r + 1 M μ M_\mu M μ μ − 1 \mu-1 μ − 1 M μ = 0 M_\mu=0 M μ = 0
Lie G 0 M = M . (16) \operatorname{Lie}_{G_0}M=M. \tag*{(16)} Lie G 0 M = M . (16) We can now determine the possible products. Their coefficients are independent of U U U λ β − k ≤ 0 \lambda_\beta-k\le0 λ β − k ≤ 0 (16) requires the coefficient of ∂ y a \partial_{y_a} ∂ y a ∂ y i ∘ ∂ y j \partial_{y_i}\circ\partial_{y_j} ∂ y i ∘ ∂ y j
1 − ω i − ω j + ω a . 1-\omega_i-\omega_j+\omega_a. 1 − ω i − ω j + ω a . For products not involving the unit, these required weights are
Every positive entry forces the corresponding coefficient to vanish. Only the ∂ W \partial_W ∂ W ∂ V i ∘ ∂ V j \partial_{V_i}\circ\partial_{V_j} ∂ V i ∘ ∂ V j λ β − k = 0 \lambda_\beta-k=0 λ β − k = 0 λ β ≤ 0 \lambda_\beta\le0 λ β ≤ 0 k ≥ 0 k\ge0 k ≥ 0 λ β = 0 \lambda_\beta=0 λ β = 0 k = 0 k=0 k = 0 G i j G_{ij} G ij g i j g_{ij} g ij G G G
Finally, let
j : = ⨁ i = 1 s O ∂ V i ⊕ O ∂ W . \mathfrak j:=\bigoplus_{i=1}^{s}\mathcal O\partial_{V_i}\oplus\mathcal O\partial_W. j := i = 1 ⨁ s O ∂ V i ⊕ O ∂ W . The displayed multiplication shows that j \mathfrak j j
j 2 = O ∂ W , j 3 = 0 , ( Der R c o n t ( O ) , ∘ ) / j ≃ O . \mathfrak j^2=\mathcal O\partial_W,\qquad\mathfrak j^3=0,\qquad\left(\operatorname{Der}^{\mathrm{cont}}_R(\mathcal O),\circ\right)/\mathfrak j\simeq\mathcal O. j 2 = O ∂ W , j 3 = 0 , ( Der R cont ( O ) , ∘ ) / j ≃ O . For the equality in the square, choose i , j i,j i , j g i j ≠ 0 g_{ij}\ne0 g ij = 0 G i j G_{ij} G ij Ω \Omega Ω j ⊗ O Ω \mathfrak j\otimes_{\mathcal O}\Omega j ⊗ O Ω Ω \Omega Ω s + 1 s+1 s + 1
Proof of Theorem 3.1
We may assume that Δ ≠ ∅ \Delta\ne\varnothing Δ = ∅ Δ S \Delta_S Δ S n : = ρ S + 2 n:=\rho_S+2 n := ρ S + 2 Δ S \Delta_S Δ S ( g i j ) (g_{ij}) ( g ij )
Ω : = Frac ( O ^ ) , \Omega:=\operatorname{Frac}(\widehat{\mathcal O}), Ω := Frac ( O ) , and denote the resulting two algebras by B + B_{+} B + B − B_{-} B −
m − = ⨁ i = 1 ρ S Ω ∂ V i ⊕ Ω ∂ W ⊂ B − \mathfrak{m}_{-}=\bigoplus_{i=1}^{\rho_{S}}\Omega\partial V_{i}\oplus\Omega\partial W\subset B_{-} m − = i = 1 ⨁ ρ S Ω ∂ V i ⊕ Ω ∂ W ⊂ B − such that
B − / m − ≃ Ω , m − 2 = Ω ∂ W , m − 3 = 0. B_{-}/\mathfrak{m}_{-}\simeq\Omega,\qquad\mathfrak{m}_{-}^{2}=\Omega\partial W,\qquad\mathfrak{m}_{-}^{3}=0. B − / m − ≃ Ω , m − 2 = Ω ∂ W , m − 3 = 0. Here the equality for the square follows from the invertibility of the matrix ( G i j ) (G_{ij}) ( G ij ) B − B_{-} B − m − \mathfrak{m}_{-} m −
dim Ω B − = n , dim Ω m − = n − 1 , dim Ω m − 2 = 1. \dim_{\Omega}B_{-}=n,\qquad\dim_{\Omega}\mathfrak{m}_{-}=n-1,\qquad\dim_{\Omega}\mathfrak{m}_{-}^{2}=1. dim Ω B − = n , dim Ω m − = n − 1 , dim Ω m − 2 = 1. It remains to check that its eigenvalue does not occur on B + B_{+} B + P ± ( T ) ∈ O ^ [ T ] P_{\pm}(T)\in\widehat{\mathcal{O}}[T] P ± ( T ) ∈ O [ T ] F 1 F^{1} F 1
P ‾ ± ( T ) = ( T − λ ± ( 0 ) ) n , λ ± ( 0 ) = 2 q f + u ′ ± 2 q f ε z / 2 . \overline{P}_{\pm}(T)=(T-\lambda_{\pm}^{(0)})^{n},\qquad\lambda_{\pm}^{(0)}=2q_{f}+u'\pm2q_{f}\varepsilon^{z/2}. P ± ( T ) = ( T − λ ± ( 0 ) ) n , λ ± ( 0 ) = 2 q f + u ′ ± 2 q f ε z /2 . Consequently,
Res T ( P ‾ + , P ‾ − ) = ± ( 4 q f ε z / 2 ) n 2 \operatorname{Res}_{T}(\overline{P}_{+},\overline{P}_{-})=\pm\left(4q_{f}\varepsilon^{z/2}\right)^{n^{2}} Res T ( P + , P − ) = ± ( 4 q f ε z /2 ) n 2 is a unit. Completeness for the filtration implies that Res T ( P + , P − ) \operatorname{Res}_{T}(P_{+},P_{-}) Res T ( P + , P − ) P + P_{+} P + P − P_{-} P − Ω \Omega Ω
Since B − B_{-} B − B − B_{-} B − B + × B − B_{+}\times B_{-} B + × B − n n n
Finally, apply § 2.3 to the injective homomorphism (8) . The subsequent formal coordinate changes are invertible. Local-factor dimensions, radical filtrations, and algebraic multiplicities are unchanged after extension to algebraically closed fraction fields. Thus
Q H e v e n ( X ) ⊗ N o v X r e d K X QH^{\mathrm{even}}(X)\otimes_{\mathrm{Nov}^{\mathrm{red}}_{X}}\mathbb{K}_{X} Q H even ( X ) ⊗ Nov X red K X has a local factor H − \mathcal{H}_{-} H − κ t ′ \kappa_{t}' κ t ′ ρ S + 2 \rho_{S}+2 ρ S + 2 ρ S ≥ 1 \rho_{S}\geq1 ρ S ≥ 1 Proposition 2.2 therefore implies that X X X
Birational reduction and irrationality
Theorem 3.1 requires Δ S = 2 K S + Δ \Delta_{S}=2K_{S}+\Delta Δ S = 2 K S + Δ ∣ Δ S ∣ ≠ ∅ |\Delta_{S}|\ne\varnothing ∣ Δ S ∣ = ∅ Δ S \Delta_{S} Δ S [16 Conjecture 1.2] :
Theorem 4.1. Let π : X → S \pi:X\to S π : X → S Δ \Delta Δ
∣ Δ S ∣ ≠ ∅ , |\Delta_{S}|\ne\varnothing, ∣ Δ S ∣ = ∅ , then X X X
First, recall from [16 Sections 3.8–3.9] that for every decomposition Δ = A + B \Delta=A+B Δ = A + B A ⋅ B A\cdot B A ⋅ B Δ \Delta Δ Γ ⊂ Δ \Gamma\subset\Delta Γ ⊂ Δ
Γ ⋅ ( Δ − Γ ) ≥ 2. (17) \Gamma\cdot(\Delta-\Gamma)\geq2. \tag*{(17)} Γ ⋅ ( Δ − Γ ) ≥ 2. (17) Indeed, this intersection number is even, and if it were zero, Γ \Gamma Γ Δ \Delta Δ
The birational construction we need is [16 Proposition 8.5] . If Γ ⊂ S \Gamma\subset S Γ ⊂ S ( − 1 ) (-1) ( − 1 )
Γ ⊄ Δ , Γ ⋅ Δ = 0 , Γ ⊄ Δ , Γ ⋅ Δ = 1 , Γ ⊂ Δ , Γ ⋅ ( Δ − Γ ) = 2 , (18) \begin{aligned}
\Gamma&\not\subset\Delta,\quad\Gamma\cdot\Delta= 0,\\
\Gamma&\not\subset\Delta,\quad\Gamma\cdot\Delta= 1,\\
\Gamma&\subset\Delta,\quad\Gamma\cdot(\Delta-\Gamma) = 2, \tag*{(18)}
\end{aligned} Γ Γ Γ ⊂ Δ , Γ ⋅ Δ = 0 , ⊂ Δ , Γ ⋅ Δ = 1 , ⊂ Δ , Γ ⋅ ( Δ − Γ ) = 2 , (18) then its contraction f : S → S ′ f:S\to S' f : S → S ′ φ : X ⇢ X ′ \varphi:X\dashrightarrow X' φ : X ⇢ X ′ π ′ : X ′ → S ′ \pi':X'\to S' π ′ : X ′ → S ′
π ′ ∘ φ = f ∘ π , Δ ′ = f ∗ Δ . \pi'\circ\varphi=f\circ\pi,\qquad\Delta'=f_*\Delta. π ′ ∘ φ = f ∘ π , Δ ′ = f ∗ Δ. The construction gives a smooth total space X ′ X' X ′ π ′ \pi' π ′ S ′ S' S ′ X ′ X' X ′
Proposition 4.2. Under the hypotheses of Theorem 4.1 , there is a birational morphism f : S → S n e f f:S\to S_{\mathrm{nef}} f : S → S nef and a projective standard conic bundle
π n e f : X n e f ⟶ S n e f \pi_{\mathrm{nef}}:X_{\mathrm{nef}}\longrightarrow S_{\mathrm{nef}} π nef : X nef ⟶ S nef with smooth total space, birational to X X X over f f f such that S n e f S_{\mathrm{nef}} S nef is a smooth projective rational surface and
Δ n e f ≠ ∅ , 2 K S n e f + Δ n e f is nef and linearly equivalent to an effective divisor. \Delta_{\mathrm{nef}}\ne\varnothing,\qquad2K_{S_{\mathrm{nef}}}+\Delta_{\mathrm{nef}}\text{ is nef and linearly equivalent to an effective divisor.} Δ nef = ∅ , 2 K S nef + Δ nef is nef and linearly equivalent to an effective divisor. Here, Δ n e f \Delta_{\mathrm{nef}} Δ nef denotes the discriminant of π n e f \pi_{\mathrm{nef}} π nef
Proof. If Δ S \Delta_S Δ S Γ \Gamma Γ Δ S ⋅ Γ < 0 \Delta_S\cdot\Gamma<0 Δ S ⋅ Γ < 0 E ∈ ∣ Δ S ∣ E\in|\Delta_S| E ∈ ∣ Δ S ∣ Γ \Gamma Γ E E E E = m Γ + E 0 E=m\Gamma+E_0 E = m Γ + E 0 m > 0 m>0 m > 0 Γ ⊄ Supp ( E 0 ) \Gamma\not\subset\operatorname{Supp}(E_0) Γ ⊂ Supp ( E 0 )
0 > E ⋅ Γ = m Γ 2 + E 0 ⋅ Γ , 0>E\cdot\Gamma=m\Gamma^2+E_0\cdot\Gamma, 0 > E ⋅ Γ = m Γ 2 + E 0 ⋅ Γ , so Γ 2 < 0 \Gamma^2<0 Γ 2 < 0
Γ 2 = − a , g = p a ( Γ ) , a ≥ 1 , \Gamma^2=-a,\qquad g=p_a(\Gamma),\qquad a\ge1, Γ 2 = − a , g = p a ( Γ ) , a ≥ 1 , where p a p_a p a K S ⋅ Γ = 2 g − 2 + a K_S\cdot\Gamma=2g-2+a K S ⋅ Γ = 2 g − 2 + a
If Γ ⊄ Δ \Gamma\not\subset\Delta Γ ⊂ Δ
0 > Δ S ⋅ Γ = 4 g − 4 + 2 a + Δ ⋅ Γ . 0>\Delta_S\cdot\Gamma=4g-4+2a+\Delta\cdot\Gamma. 0 > Δ S ⋅ Γ = 4 g − 4 + 2 a + Δ ⋅ Γ. Since g ≥ 0 g\ge0 g ≥ 0 Δ ⋅ Γ ≥ 0 \Delta\cdot\Gamma\ge0 Δ ⋅ Γ ≥ 0 g = 0 g=0 g = 0 a = 1 a=1 a = 1 Δ ⋅ Γ ∈ { 0 , 1 } \Delta\cdot\Gamma\in\{0,1\} Δ ⋅ Γ ∈ { 0 , 1 }
If Γ ⊂ Δ \Gamma\subset\Delta Γ ⊂ Δ b : = Γ ⋅ ( Δ − Γ ) ≥ 0 b:=\Gamma\cdot(\Delta-\Gamma)\ge0 b := Γ ⋅ ( Δ − Γ ) ≥ 0
0 > Δ S ⋅ Γ = 4 g − 4 + a + b . 0>\Delta_S\cdot\Gamma=4g-4+a+b. 0 > Δ S ⋅ Γ = 4 g − 4 + a + b . Thus, g = 0 g=0 g = 0 Γ \Gamma Γ (17) , b ≥ 2 b\ge2 b ≥ 2 a + b < 4 a+b<4 a + b < 4 a = 1 a=1 a = 1 b = 2 b=2 b = 2 Γ \Gamma Γ ( − 1 ) (-1) ( − 1 ) (18) .
Let f : S → S ′ f:S\to S' f : S → S ′ Γ \Gamma Γ [16 Proposition 8.5] . Put Δ S ′ : = 2 K S ′ + Δ ′ \Delta_{S'}:=2K_{S'}+\Delta' Δ S ′ := 2 K S ′ + Δ ′ K S ∼ f ∗ K S ′ + Γ K_S\sim f^*K_{S'}+\Gamma K S ∼ f ∗ K S ′ + Γ
Δ S ∼ f ∗ Δ S ′ + { 2 Γ , Γ ⊄ Δ , Γ ⋅ Δ = 0 , Γ , in the other two cases. (19) \Delta_S\sim f^*\Delta_{S'}+
\begin{cases}
2\Gamma, & \Gamma\not\subset\Delta,\ \Gamma\cdot\Delta=0,\\
\Gamma, & \text{in the other two cases.}
\end{cases} \tag*{(19)} Δ S ∼ f ∗ Δ S ′ + { 2Γ , Γ , Γ ⊂ Δ , Γ ⋅ Δ = 0 , in the other two cases. (19) To see this, f ∗ Δ ′ = Δ f^*\Delta'=\Delta f ∗ Δ ′ = Δ f ∗ Δ ′ = Δ + Γ f^*\Delta'=\Delta+\Gamma f ∗ Δ ′ = Δ + Γ Δ − Γ \Delta-\Gamma Δ − Γ Γ \Gamma Γ Δ \Delta Δ f ( Γ ) f(\Gamma) f ( Γ )
f ∗ Δ ′ = ( Δ − Γ ) + 2 Γ = Δ + Γ . f^*\Delta'=(\Delta-\Gamma)+2\Gamma=\Delta+\Gamma. f ∗ Δ ′ = ( Δ − Γ ) + 2Γ = Δ + Γ. Pushing forward an effective representative of Δ S \Delta_S Δ S ∣ Δ S ′ ∣ ≠ ∅ |\Delta_{S'}|\ne\varnothing ∣ Δ S ′ ∣ = ∅
Repeat this construction whenever the resulting adjoint class is not nef. Effectivity is preserved at each step, and every contraction lowers the Picard number of the base by one. The process therefore terminates at a smooth projective rational surface S n e f S_{\mathrm{nef}} S nef X X X ∣ 2 K S n e f ∣ ≠ ∅ \lvert2K_{S_{\mathrm{nef}}}\rvert\ne\varnothing ∣ 2 K S nef ∣ = ∅
H 0 ( S n e f , O S n e f ( 2 K S n e f ) ) = 0 , H^{0}\left(S_{\mathrm{nef}},\mathcal{O}_{S_{\mathrm{nef}}}\left(2K_{S_{\mathrm{nef}}}\right)\right)=0, H 0 ( S nef , O S nef ( 2 K S nef ) ) = 0 , since every smooth projective rational surface has vanishing positive plurigenera. □
Proof of Theorem 4.1 . Proposition 4.2 gives a projective standard conic bundle π n e f : X n e f → S n e f \pi_{\mathrm{nef}}: X_{\mathrm{nef}} \to S_{\mathrm{nef}} π nef : X nef → S nef 2 K S n e f + Δ n e f 2K_{S_{\mathrm{nef}}}+\Delta_{\mathrm{nef}} 2 K S nef + Δ nef Theorem 3.1 implies that X n e f X_{\mathrm{nef}} X nef X X X X n e f X_{\mathrm{nef}} X nef
Let J ( X ) J(X) J ( X ) X X X
J ( X ) : = H 3 ( X , C ) F 2 H 3 ( X , C ) + im H 3 ( X , Z ) , F 2 H 3 ( X , C ) : = H 3 , 0 ( X ) ⊕ H 2 , 1 ( X ) . J(X):=\frac{H^{3}(X,\mathbb{C})}{F^{2}H^{3}(X,\mathbb{C})+\operatorname{im}H^{3}(X,\mathbb{Z})},\qquad F^{2}H^{3}(X,\mathbb{C}):=H^{3,0}(X)\oplus H^{2,1}(X). J ( X ) := F 2 H 3 ( X , C ) + im H 3 ( X , Z ) H 3 ( X , C ) , F 2 H 3 ( X , C ) := H 3 , 0 ( X ) ⊕ H 2 , 1 ( X ) . For the conic bundles over rational surfaces considered here, this is a principally polarized abelian variety. All isomorphisms below preserve the principal polarizations.
Corollary 4.3. Let π : X → S \pi:X\to S π : X → S Δ \Delta Δ X X X ∣ 2 K S + Δ ∣ = ∅ \lvert2K_{S}+\Delta\rvert=\varnothing ∣ 2 K S + Δ ∣ = ∅ J ( X ) J(X) J ( X )
Proof. If X X X Theorem 4.1 gives ∣ 2 K S + Δ ∣ = ∅ \lvert2K_{S}+\Delta\rvert=\varnothing ∣ 2 K S + Δ ∣ = ∅ J ( X ) J(X) J ( X ) [16 Corollary 7.3] .
Conversely, assume both conditions. If Δ = ∅ \Delta=\varnothing Δ = ∅ Br ( S ) \operatorname{Br}(S) Br ( S ) X X X S × P 1 S\times\mathbb{P}^{1} S × P 1 [16 Lemma 3.7] .
Suppose now that Δ ≠ ∅ \Delta\ne\varnothing Δ = ∅ [16 Proposition 8.6] , X → S X\to S X → S X ′ → S ′ X'\to S' X ′ → S ′ Δ ′ \Delta' Δ ′ S ′ S' S ′ Δ ′ ⋅ F ≤ 3 \Delta'\cdot F\le3 Δ ′ ⋅ F ≤ 3 F F F S ′ ≃ P 2 S'\simeq\mathbb{P}^{2} S ′ ≃ P 2 X ′ X' X ′ [16 Proposition 5.6] . In the plane case, J ( X ′ ) J(X') J ( X ′ ) [16 Proposition 7.2] . Thus [16 Theorem 9.1] implies that X ′ X' X ′ X X X X ′ X' X ′
Nodal quartic double solids
We now apply the conic-bundle calculation to double covers of P 3 \mathbb{P}^{3} P 3 nodal quartic double solid we mean a double cover τ : Y → P 3 \tau:Y\to\mathbb{P}^{3} τ : Y → P 3 B ⊂ P 3 B\subset\mathbb{P}^{3} B ⊂ P 3 Y Y Y B B B N N N Y Y Y Q \mathbb{Q} Q Y Y Y
We first construct a smooth standard conic-bundle model in the Q \mathbb{Q} Q Q \mathbb{Q} Q B B B
The standard model and its quantum cohomology
The geometric input for the quantum calculation is the following standard-model construction.
Proposition 5.1 ([7 Theorem 4.2] ). Let Y Y Y Q \mathbb{Q} Q N ≥ 1 N \ge1 N ≥ 1 π : X → S \pi: X \to S π : X → S X X X Y Y Y S → P 2 S \to\mathbb{P}^{2} S → P 2 N − 1 N-1 N − 1
Δ ∼ − 2 K S . \Delta\sim-2K_{S}. Δ ∼ − 2 K S . In particular,
ρ S = N , K S 2 = 10 − N , Δ S ∼ 0. \rho_{S}=N,\qquad K_{S}^{2}=10-N,\qquad\Delta_{S}\sim0. ρ S = N , K S 2 = 10 − N , Δ S ∼ 0. Corollary 5.2. Every Q \mathbb{Q} Q N ≥ 1 N \ge1 N ≥ 1 X X X Proposition 5.1 has a local Artinian factor H − \mathcal{H}_{-} H −
Q H e v e n ( X ) ⊗ Nov X r e d K X QH^{\mathrm{even}}(X)\otimes\operatorname{Nov}^{\mathrm{red}}_{X}\mathbb{K}_{X} Q H even ( X ) ⊗ Nov X red K X satisfying
dim K X H − = N + 2 , dim K X rad ( H − ) = N + 1 , \dim_{\mathbb{K}_{X}}\mathcal{H}_{-}=N+2,\qquad\dim_{\mathbb{K}_{X}}\operatorname{rad}(\mathcal{H}_{-})=N+1, dim K X H − = N + 2 , dim K X rad ( H − ) = N + 1 , dim K X ( rad ( H − ) 2 ) = 1 , rad ( H − ) 3 = 0. \dim_{\mathbb{K}_{X}}\left(\operatorname{rad}(\mathcal{H}_{-})^{2}\right)=1,\qquad\operatorname{rad}(\mathcal{H}_{-})^{3}=0. dim K X ( rad ( H − ) 2 ) = 1 , rad ( H − ) 3 = 0. The operator κ t ′ \kappa_{t'} κ t ′ X X X
Proof. For N ≥ 1 N \ge1 N ≥ 1 Proposition 5.1 gives Δ S ∼ 0 \Delta_{S}\sim0 Δ S ∼ 0 ρ S = N \rho_{S}=N ρ S = N Δ \Delta Δ 2 K S ∼ 0 2K_{S}\sim0 2 K S ∼ 0 S S S Theorem 3.1 therefore gives the asserted local factor and multiplicity. Since N + 2 ≥ 3 N+2\ge3 N + 2 ≥ 3 Proposition 2.2 excludes rationality of X X X Y Y Y N = 0 N=0 N = 0 [7 Theorem 1.2 and the preceding discussion] . □
Rationality and the configuration of nodes
The preceding corollary settles the Q \mathbb{Q} Q
Theorem 5.3. Let Y Y Y N N N
Y is rational ⟺ Y is not Q -factorial and N ≠ 6. Y\text{ is rational}\quad\Longleftrightarrow\quad Y\text{ is not }\mathbb{Q}\text{-factorial and }N\ne6. Y is rational ⟺ Y is not Q -factorial and N = 6. Proof. If Y Y Y Corollary 5.2 shows that it is not Q \mathbb{Q} Q [7 Theorem 1.2] excludes N = 6 N=6 N = 6 Y Y Y Q \mathbb{Q} Q N ≠ 6 N\ne6 N = 6 [7 Theorem 1.5] . □
The criterion can also be read directly from the nodes of the branch surface. Put Σ : = Sing ( B ) \Sigma:= \operatorname{Sing}(B) Σ := Sing ( B ) [7 Theorem 1.6] , Y Y Y Q \mathbb{Q} Q
ev Σ : H 0 ( P 3 , O P 3 ( 2 ) ) ⟶ H 0 ( Σ , O Σ ( 2 ) ) \operatorname{ev}_{\Sigma}: H^{0}(\mathbb{P}^{3},\mathcal{O}_{\mathbb{P}^{3}}(2))\longrightarrow H^{0}(\Sigma,\mathcal{O}_{\Sigma}(2)) ev Σ : H 0 ( P 3 , O P 3 ( 2 )) ⟶ H 0 ( Σ , O Σ ( 2 )) is surjective. Write the nodes as P j = [ z j 0 : z j 1 : z j 2 : z j 3 ] P_{j}=[z_{j0}:z_{j1}:z_{j2}:z_{j3}] P j = [ z j 0 : z j 1 : z j 2 : z j 3 ] ( z j 0 , … , z j 3 ) (z_{j0},\ldots,z_{j3}) ( z j 0 , … , z j 3 ) z a z b z_{a}z_{b} z a z b 0 ≤ a ≤ b ≤ 3 0\le a\le b\le3 0 ≤ a ≤ b ≤ 3 ev Σ \operatorname{ev}_{\Sigma} ev Σ N × 10 N\times10 N × 10
( M Σ ) j , ( a , b ) : = z j a z j b . (M_{\Sigma})_{j,(a,b)}:=z_{ja}z_{jb}. ( M Σ ) j , ( a , b ) := z j a z j b . Changing a representative rescales the corresponding row by a nonzero scalar, so the rank is independent of these choices. Theorem 5.3 therefore gives
Y is rational ⟺ rank C M Σ < N and N ≠ 6. Y\text{ is rational}\quad\Longleftrightarrow\quad\operatorname{rank}_{\mathbb{C}}M_{\Sigma}<N\quad\text{and}\quad N\ne6. Y is rational ⟺ rank C M Σ < N and N = 6. In particular, every nodal quartic double solid with N ≥ 11 N\ge11 N ≥ 11 M Σ M_{\Sigma} M Σ
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
Email address: leonardofcavenaghi@gmail.com
Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794, USA. & International Center for Mathematical Sciences (ICMS), Sofia, Bulgaria. & Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria.
Email address: lkatzarkov@gmail.com
UMASS BOSTON, 100 Morrissey Blvd., Boston, MA 02125
Email address: daniel.pomerleano@umb.edu
Department of Mathematics, Columbia University, New York, NY 10027, USA
Email address: zhang.zhijia@columbia.edu
