Mukai's program for curves on a K3 surface

Arbarello, Enrico; Bruno, Andrea; Sernesi, Edoardo

doi:10.14231/ag-2014-023

Cited by 21 publications

(65 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will therefore first develop some auxiliary results on E 2k+2,2 . Recall that by [7] these spaces are irreducible and that H ∼ kE 1…”

Section: A Technical Resultsmentioning

confidence: 99%

“…We find α = 0 as above. We have 2F 1 (20)). It follows that h i (2 F 1 +2 F 2 − H) = 0 for i = 0, 1 and h 0 (4 F 1 +4 F 2 − H) = 2, whence β = 2 by (14).…”

Section: Fiber Dimensions Of Moduli Mapsmentioning

confidence: 99%

“…Notable are the relations of pathologies of c g with the existence of Fano and Mukai manifolds [9,6]. Also recall that Mukai's program towards reconstructing a fiber of c g is now proven [33,1,19], and that the image of c g has been recently characterized, via the Gauss-Wahl map, for Brill-Noether-Petri general curves [2,45].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Moduli of curves on Enriques surfaces

Ciliberto¹,

Dedieu²,

Galati³

et al. 2020

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…We will therefore first develop some auxiliary results on E 2k+2,2 . Recall that by [7] these spaces are irreducible and that H ∼ kE 1…”

Section: A Technical Resultsmentioning

confidence: 99%

“…We find α = 0 as above. We have 2F 1 (20)). It follows that h i (2 F 1 +2 F 2 − H) = 0 for i = 0, 1 and h 0 (4 F 1 +4 F 2 − H) = 2, whence β = 2 by (14).…”

Section: Fiber Dimensions Of Moduli Mapsmentioning

confidence: 99%

See 1 more Smart Citation

Moduli of curves on Enriques surfaces

Ciliberto¹,

Dedieu²,

Galati³

et al. 2020

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…Using that X has Picard number 1 and B 2 is 20 or at least 24, Arbarello-Bruno-Sernesi and Feyzbakhsh showed that C lies on a unique K3 surface of Picard number 1 [2],…”

Section: Bott Vanishing For Ksurfaces Of Picard Numbermentioning

confidence: 99%

Bott vanishing for algebraic surfaces

Totaro

2020

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

For William Fulton on his eightieth birthdayA smooth projective variety X over a field is said to satisfy Bott vanishing if H j (X, Ω i X ⊗ L) = 0 for all ample line bundles L, all i ≥ 0, and all j > 0. Bott proved this when X is projective space. Danilov and Steenbrink extended Bott vanishing to all smooth projective toric varieties; proofs can be found in [4,7,28,15]. What does Bott vanishing mean? It does not have a clear geometric interpretation in terms of the classification of algebraic varieties. But it is useful when it holds, as a sort of preprocessing step, since the vanishing of higher cohomology lets us compute the spaces of sections of various important vector bundles. Bott vanishing includes Kodaira vanishing as a special case (where i equals n := dim X), but it says much more.For example, any Fano variety that satisfies Bott vanishing must be rigid, since H 1 (X, T X) = H 1 (X, Ω n−1 X ⊗ K * X ) = 0 for X Fano. So Bott vanishing holds for only finitely many smooth complex Fano varieties in each dimension. Even among rigid Fano varieties, Bott vanishing fails for quadrics of dimension at least 3 and for Grassmannians other than projective space [7, section 4]. As a result, Achinger, Witaszek, and Zdanowicz asked whether a rationally connected variety that satisfies Bott vanishing must be a toric variety [1, after Theorem 4].In this paper, we exhibit several new classes of varieties that satisfy Bott vanishing. First, we answer Achinger-Witaszek-Zdanowicz's question: there are non-toric rationally connected varieties that satisfy Bott vanishing, since Bott vanishing holds for the quintic del Pezzo surface (Theorem 2.1). Over an algebraically closed field, a quintic del Pezzo surface is isomorphic to the moduli space M 0,5 of 5-pointed stable curves of genus zero. It is the only rigid del Pezzo surface that is not toric: del Pezzo surfaces of degree at least 5 are rigid, and those of degree at least 6 are toric. (The quintic del Pezzo surface also does not have a lift of the Frobenius endomorphism from Z/p to Z/p 2 , a property known to imply Bott vanishing [7], [1, Proposition 7.1.4].) In view of this example, there is a good hope of finding more Fano or rationally connected varieties that satisfy Bott vanishing.We also consider varieties that are not rationally connected, with most of the paper devoted to K3 surfaces. Bott vanishing holds for abelian varieties over any field: it reduces to Kodaira vanishing, since the tangent bundle is trivial. On the other hand, Riemann-Roch shows that Bott vanishing fails for all K3 surfaces of degree less than 20 (Theorem 3.1). But recent work of Ciliberto-Dedieu-Sernesi and Feyzbakhsh [9,14] implies: Bott vanishing holds for all K3 surfaces of degree 20 or at least 24 with Picard number 1 (Theorem 3.2). Version 2 of this paper on the arXiv gave a more elementary proof, not using Feyzbakhsh's work on Mukai's program (reconstructing a K3 surface from a curve), but here we give a short proof using her work. Surprisingly, Bott vanishing fails in degree 22.More stron...

show abstract

“…For the other compact examples, taking their dual fibrations seems to produce new examples of Lagrangian fibrations, making this a potentially fertile direction to explore. 2 The GL-Hitchin system…”

Section: Introductionmentioning

confidence: 99%

On Lagrangian fibrations by Jacobians, II

Sawon

2015

Commun. Contemp. Math.

View full text Add to dashboard Cite

We survey Lagrangian fibrations of holomorphic symplectic varieties, both compact and non-compact, whose fibres are Jacobians and Prym varieties. THE GL-HITCHIN SYSTEMThe degeneration of Donagi, Ein, and Lazarsfeld can be generalized to some Lagrangian fibrations by Prym varieties, giving a very concrete connection between certain compact and noncompact examples. In other cases, we can detect some analogies between compact and non-compact examples without there being a clear connection. Still, there remain many non-compact Lagrangian fibrations which do not have obvious compact counterparts (and conversely).Here is a summary of the paper. In Section 2 we describe the GL-Hitchin system, a noncompact Lagrangian fibration by Jacobians of curves, and how it is related to its dual fibration. In Section 3 we consider compact Lagrangian fibrations, particularly the Beauville-Mukai system. Again we describe how it is related to its dual fibration, we describe the Donagi-Ein-Lazarsfeld degeneration to a compactification of the GL-Hitchin system, and we summarize what is known about the classification of Lagrangian fibrations by Jacobians of curves. In Section 4 we describe the other Hitchin systems, which are fibred by Prym varieties, and the duality relations between them. In Section 5 we turn to compact Lagrangian fibrations by Prym varieties. We describe the Markushevich-Tikhomirov system, its dual fibration, the Matteini system, and some other examples. In Section 6 we mention a few original results of the author (some obtained jointly with Chen Shen): we extend the Donagi-Ein-Lazarsfeld degeneration result to Lagrangian fibrations by Prym varieties, we speculate on the dual fibration of the Matteini system, and we describe the dual fibration of the Debarre system. Finally, in Section 7 we summarize all of the known examples of Lagrangian fibrations by Jacobians and Prym varieties in Table 1; this suggests where we should perhaps search for new examples.This work was presented at the satellite meeting of the ICM 2018 "Moduli spaces in algebraic geometry and applications" held in Campinas, 26-31 July, 2018. The author is grateful to the organizers for the invitation to speak at this meeting, and also grateful for support from the National Science Foundation, award DMS-1555206.Theorem 1 (Hitchin [20]) For coprime n and d, M GL is a holomorphic symplectic manifold, i.e., it admits a natural holomorphic symplectic form σ. In fact, M GL admits a hyperkähler metric. The Hitchin system3 Remark Denote by Bun GL the moduli space of stable bundles on Σ of rank n and degree d. If E ∈ Bun GL , then the inequality above is satisfied for all subbundles F . Therefore we could choose any Higgs field Φ, and the pair (E, Φ) would be a stable Higgs bundle. Consider the cotangent space to Bun GL at E. By Serre dualitySo in this case (E, Φ) is really a point of the cotangent bundle T * Bun GL . This shows that T * Bun GL ⊂ M GL , and in fact it is a dense open subset. The holomorphic symplectic form σ on M GL is an extension of the canoni...

show abstract

Mukai's program for curves on a K3 surface

Cited by 21 publications

References 12 publications

Moduli of curves on Enriques surfaces

Moduli of curves on Enriques surfaces

Bott vanishing for algebraic surfaces

On Lagrangian fibrations by Jacobians, II

Contact Info

Product

Resources

About