Related papers: Secant loci on moduli of Prym varieties

Let (S,H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c_1(E),H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether…

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…

We study moduli spaces of K3 surfaces endowed with a Nikulin involution and their image in the moduli space R_g of Prym curves of genus g. We observe a striking analogy with Mukai's well-known work on ordinary K3 surfaces. Many of Mukai's…

We prove the irreducibility of the moduli space of rank 2 semistable torsion free sheaves (with a generic polarization and any value of c_2) on a K3 or a del Pezzo surface. In the case of a K3 surface, we need to prove a result on the…

We survey basic results concerning Prym varieties, the Prym-Brill-Noether theory initiated by Welters, and Brill-Noether theory of general \'etale double covers of curves of genus g>=2. We then specialize to curves on Nikulin surfaces and…

Let (S,H) be a polarized K3 surface, $E$ be a coherent sheaf on S and W be a linear subspace in the space of global sections H^0(S,E). If we are lucky, there is an exact sequence 0 -> W tensor O -> E -> E' -> 0, which gives a correspondence…

We prove that the Jacobian of a general curve C of genus g=2a+1, with g>4, can be realized as a Prym-Tyurin variety for the Brill-Noether curve W^1_{a+2}(C). As a consequence of this result we are able to compute the class of the sum of the…

Lazarsfeld proved Brill--Noether generality of any smooth curve in the linear system $|H|$ where $(X,H)$ is a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb{Z}\cdot H$. Mukai introduced the notion of Brill--Noether generality for…

For $r\geq 3$ and $g= \frac{r(r+1)}{2}$, we study the Prym-Brill-Noether variety $V^r(C,\eta)$ associated to Prym curves $[C,\eta]$. The locus $\mathcal{R}_g^r$ in $\mathcal{R}_g$ parametrizing Prym curves $(C, \eta)$ with nonempty…

We study some natural linear systems carried by polarized Nikulin surfaces of genus g. We determine their positivity and establish their Brill-Noether theory. As an application, we compute the class of some natural effective divisors…

Primitively polarized genus $g$ Nikulin surfaces $(S,M,H)$ are of two types, that we call standard and non-standard depending on whether the lattice embedding $\mathbb{Z}[H] \oplus_{\perp} \mathbf{N} \subset \rm{Pic}(S)$ is primitive. Here…

Mukai showed that projective models of Brill-Noether general polarized K3 surfaces of genus $6-10$ and $12$ are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill-Noether general…

In this paper, we study the Brill-Noether theory of the normalizations of singular, irreducible curves on a $K3$ surface. We introduce a {\em singular} Brill-Noether number $\rho_{sing}$ and show that if the Picard group of the K3 surface…

The paper is devoted to highlighting several novel aspects of the moduli space of curves of genus 13, the first genus g where phenomena related to K3 surfaces no longer govern the birational geometry of M_g. We compute the class of the…

Let C be a smooth projective complex curve of genus $g\geq2$. We investigate the Brill-Noether locus consisting of stable bundles of rank 2 and canonical determinant having at least $k$ independent sections. Using the Hecke correpondence we…

We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more…

A refined Brill--Noether theory seeks to determine which linear series are admitted by a ``general'' curve in a particular Brill--Noether locus. However, as Brill--Noether loci are not irreducible in general, a coarse answer is given by the…

Generalizing the well-known Green Conjecture on syzygies of canonical curves, Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy if…

We explicitly construct Brill--Noether general $K3$ surfaces of genus $4,6$ and $8$ having the maximal number of elliptic pencils of degrees $3, 4$ and $5$, respectively, and study their moduli spaces and moduli maps to the moduli space of…

We extend a previous result of Feyzbakhsh concerning the injectivity of a map of moduli spaces and we use this result to construct curves whose Brill-Noether loci have unexpected dimension.