Abstract. Severi varieties and Brill-Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface S with polarization L of type (1, n), we prove nonemptiness and regularity of the Severi variety parametrizing δ-nodal curves in the linear system |L| for 0 ≤ δ ≤ n−1 = p −2 (here p is the arithmetic genus of any curve in |L|). We also show that a general genus g curve having as nodal model a hyperplane section of some (1, n)-polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many (1, n)-polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus g curve in S equigenerically to a nodal curve. The rest of the paper deals with the BrillNoether theory of curves in |L|. It turns out that a general curve in |L| is Brill-Noether general. However, as soon as the Brill-Noether number is negative and some other inequalities are satisfied, the locus |L|

ABSTRACT. Given an ample line bundle L on a K3 surface S, we study the slope stability with respect to L of rank-3 Lazarsfeld-Mukai bundles associated with complete, base point free nets of type g 2 d on curves C in the linear system |L|. When d is large enough and C is general, we obtain a dimensional statement for the variety W 2 d (C). If the Brill-Noether number is negative, we prove that any g 2 d on any smooth, irreducible curve in |L| is contained in a g r e which is induced from a line bundle on S, thus answering a conjecture of Donagi and Morrison. Applications towards transversality of Brill-Noether loci and higher rank Brill-Noether theory are then discussed.

Green's conjecture is proved for smooth curves C lying on a rational surface S with an anticanonical pencil, under some mild hypotheses on the line bundle L = OS(C). Constancy of Clifford dimension, Clifford index and gonality of curves in the linear system |L| is also obtained.

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...

The Gieseker-Petri locus G P g is defined as the locus inside M g consisting of curves which violate the Gieseker-Petri Theorem. It is known that G P g has always some divisorial components and it has been conjectured that it is of pure codimension 1 inside M g . We prove that this holds true for genus up to 13.