Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds

Abstract

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...

Constant cycle and co-isotropic subvarieties in a Mukai system

“…Example 4.17 This example is taken from Knutsen et al [19]. As in (2.2), we consider the locus V(j, |H|) ⊂ |H| of curves C with g( C) ≤ j for j = 0, 1, 2 .…”

“…We let C j → V(2 − j, |H|) be the respective restriction of the universal curve C |H| → |H| . For 2 − j ≤ i ≤ 4 , consider the diagram in which the lower horizontal map turns out to be generically injective [19,Theorem 6.4]. Hence, f j i is generically finite.…”

Curve classes on irreducible holomorphic symplectic varieties

“…By construction, the curve R deforms in a family of dimension 2n−2. Moreover, the incidence correspondence (4) can be used to prove that all deformations of these rational curves on the Hilbert scheme of points on a general K3 (or a generalized Kummer) are actually induced by linear series on different curves on S (see [KLM,Proposition 5.3] or [KLM2,Proposition 3.6] for a proof of this).…”

“…As γ is effective and of negative square in a Picard rank one manifold, the divisor class D is a wall divisor (or MBM class) in the sense of [Mo,Definition 1.2]. Being a wall divisor is preserved by deformations in the Hodge locus of γ by [Mo, Theorem 1.3], so we can take a projective small deformation Y ′ of Y such that γ is contracted by a map Y ′ → X ′ by [KLM2,Theorem 2.5]. Now, this implies that there exists a map φ : Y → X to a singular symplectic manifold X which contracts γ without taking any deformation (see [BL,Theorem 1.1]).…”

Hyperbolic geometry of the ample cone of a hyperkähler manifold

“…Geometrically, the MBM classes are characterized among negative integral (1, 1)-classes as those which are, up to a scalar multiple, represented by minimal rational curves on deformations of M under the identification of H 2 (M, Q) with H 2 (M, Q) given by the BBF form [2,3,13].…”