Severi varieties and Brill–Noether theory of curves on abelian surfaces

Knutsen, Andreas Leopold; Lelli-Chiesa, Margherita; Mongardi, Giovanni

doi:10.1515/crelle-2016-0029

Cited by 20 publications

(27 citation statements)

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“…The fact that for general t ∈ T we have dim(W 1 k+1 ( C t )) = ρ(g, 1, k + 1) follows e.g. from [KLM,Theorem 1.6,(ii)]. Notice however that here we need only the inequality dim(W 1 k+1 ( C t )) ≥ ρ(g, 1, k + 1) which follows from the positivity of ρ(g, 1, k + 1) and from the classical Brill-Noether theory (see e.g.…”

Section: 2mentioning

confidence: 96%

Polarized Parallel Transport and Uniruled Divisors on Deformations of Generalized Kummer Varieties

Mongardi

Pacienza

2017

International Mathematics Research Notices

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Section: 2mentioning

confidence: 96%

Polarized Parallel Transport and Uniruled Divisors on Deformations of Generalized Kummer Varieties

Mongardi

Pacienza

2017

International Mathematics Research Notices

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“…(ii) The condition (12) is also necessary for the existence of a curve in {L} with partial normalization of arithmetic genus g := p − δ carrying a g 1 k+ε . This follows from [KLM,Thm. 5.9 and Rem.…”

Section: Curves On Symplectic Surfaces and Their Pencilsmentioning

confidence: 84%

“…Any small deformation X t of X 0 = S [k] ε keeping the class of the rational curves algebraic contains a (2k − 2)-dimensional family of rational curves that are deformations of the rational curves in S Proof. Any irreducible family of rational curves in S [k] ε containing our family yields, by the incidence (11), a family of pairs (C, g) with C ∈ {L} and g a linear series of type g 1 k+ε on the normalization of C. By [KLM,Thm. 5.3], the rational curves will therefore move in a family of dimension precisely 2k − 2, which is the expected dimension of any family of rational curves on a (2k)-dimensional IHS manifold [Ra,Cor.…”

Section: Curves On Symplectic Surfaces and Their Pencilsmentioning

confidence: 99%

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Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds

Knutsen

Lelli-Chiesa

Mongardi

2018

Trans. Amer. Math. Soc.

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Abstract. Rational curves on Hilbert schemes of points on K3 surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up to isometry, as dual divisors to such rational curves. The locus covered by the rational curves is then described, thus exhibiting algebraically coisotropic subvarieties. This provides strong evidence for a conjecture by Voisin concerning the Chow ring of irreducible holomorphic symplectic manifolds. Some general results concerning the birational geometry of irreducible holomorphic symplectic manifolds are also proved, such as a nonprojective contractibility criterion for wall divisors. IntroductionRational curves play a pivotal role in the study of the birational geometry and the Chow ring of algebraic varieties. The present paper concerns a specific class of varieties, namely, irreducible holomorphic symplectic (IHS) manifolds and, more precisely, Hilbert schemes of points on K3 surfaces and generalised Kummer manifolds (cf. §1), and is focused on some special rational curves arising from the Brill-Noether theory of normalisations of curves lying on K3 and abelian surfaces. In order to treat the two cases simultaneously, we introduce the following notation: we set ε = 0 (respectively, ε = 1) when S is a K3 (resp., abelian) surface, and we denote by S [k] ε the Hilbert scheme of k points on S when ε = 0 and the 2k-dimensional generalised Kummer variety on S when ε = 1.In the last few years, some classical results concerning (−2)-curves on K3 surfaces have been generalised to higher dimension and in particular it was shown that rational curves fully control the birational geometry of IHS manifolds. More precisely, Ran [Ra] proved that extremal rational curves can be deformed together with the ambient IHS manifold, and this was exploited by Bayer, Hassett and Tschinkel [BHT] in order to determine the structure of the ample cone. The same result was independently obtained by the third named author [Mo1] using intrinsic properties of IHS manifolds and a deformation invariant class of divisors, the so-called wall divisors (cf. Definition 2.2), which contains all divisors dual to extremal rays. and the space of stability conditions can be used towards computing their ample cones.In this paper we use Brill-Noether theory of nodal curves on abelian and K3 surfaces in order to exhibit rational curves in S [k] ε and describe, in many cases, the locus they cover. Our construction proceeds as follows. Let (S, L) be a general primitively polarized K3 or abelian surface of genus p := p a (L) and let C ∈ |L| be a δ-nodal curve whose normalization C has a linear series of type g 1 k+ε . Existence of a family of such curves having the expected dimension (and satisfying certain additional properties) has been proved in [CK, KLM] under suitable conditions on the triple (p, k, δ), cf. Theorem 3.1. Any pencil of degree k + ε on C defines a rational curve in S [k] ε , whose cla...

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“…[17, Theorem 5.3 and Remark 5.6]). Let (S, L) be such that Pic(S) = Z[L], and V ⊂ V L g a non-empty reduced scheme.…”

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confidence: 99%

On the irreducibility of Severi varieties on 𝐾3 surfaces

Ciliberto

Dedieu

2019

Proc. Amer. Math. Soc.

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Let (S, L) be a polarized K3 surface of genus p 11 such that Pic(S) = Z[L], and δ a non-negative integer. We prove that if p 4δ − 3, then the Severi variety of δ-nodal curves in |L| is irreducible.1 Actually, the assumption in [8, Proposition 4.8] is that (S, L) be very general; it is straightforward to check that the condition Pic(S) = L is indeed sufficient for the proof in [8].2 in the sense that for all d ′ of cardinality δ, the sheets Vd and V d ′ intersect exactly along the local sheet V d∪d ′ of V L,|d∪d ′ | at C, and their respective tangent spaces at C intersect exactly along the tangent space of V d∪d ′ at C.

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Severi varieties and Brill–Noether theory of curves on abelian surfaces

Cited by 20 publications

References 43 publications

Polarized Parallel Transport and Uniruled Divisors on Deformations of Generalized Kummer Varieties

Polarized Parallel Transport and Uniruled Divisors on Deformations of Generalized Kummer Varieties

Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds

On the irreducibility of Severi varieties on 𝐾3 surfaces

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