Related papers: Maximal Brill-Noether loci via K3 surfaces

In this note we give an easy proof of the existence of generically smooth components of the expected dimension of certain Brill--Noether loci of stable rank 2 vector bundles on a curve with general moduli, with related applications to…

Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C \to \mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main…

Let $X$ be a non-singular projective curve of genus $g\ge2$ over an algebraically closed field of characteristic zero. Let $\mo$ denote the moduli space of stable bundles of rank $n$ and degree $d$ on $X$ and $\wn $ the Brill-Noether loci…

Let $(X,H)$ be a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb Z H$, and let $C\in |H|$ be a smooth curve of genus $g$. We give an upper bound on the dimension of global sections of a semistable vector bundle on $C$. This allows us…

We study the Brill-Noether stratification of the coarse moduli space of locally free stable and flat sheaves of a compact Kahler manifold, proving that these strata have quadratic algebraic singularities.

Let $C$ be a curve of genus $g\geq 2$. A coherent system on $C$ consists of a pair $(E,V)$ where $E$ is an algebraic vector bundle of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of sections of $E$. The stability of the…

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…

We study the variation of linear sections of hypersurfaces in $\mathbb{P}^n$. We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family…

This paper gives an overview of the main results of Brill-Noether Theory for vector bundles on algebraic curves.

New local and global non-abelian zeta functions for elliptic curves are studied using certain refined Brill-Noether loci in moduli spaces of semi-stable bundles. Examples of these zeta functions and a justification of using only semi-stable…

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 show that the Brill-Noether locus M^3_{18,16} is an irreducible component of the Gieseker-Petri locus in genus 18 having codimension 2 in the moduli space of curves. This result disproves a conjecture predicting that the Gieseker-Petri…

Let $V$ be a vector bundle over a smooth curve $C$. In this paper, we study twisted Brill--Noether loci parametrising stable bundles $E$ of rank $n$ and degree $e$ with the property that $h^0 (C, V \otimes E) \ge k$. We prove that, under…

We completely describe all Brill-Noether loci on metric graphs consisting of a chain of g cycles with arbitrary edge lengths, generalizing work of Cools, Draisma, Payne, and Robeva. The structure of these loci is determined by displacement…

A Brill-Noether degeneracy locus is closure in $\Pic^d(C)$ of the locus of line bundles with a specified rank function $r(a,b) = h^0(C,L(-ap-bq))$. These loci generalize the classical Brill-Noether loci $W^r_d(C)$ as well as Brill-Noether…

We study the moduli space of solutions to the Seiberg-Witten equations with $N$ spinors on a compact Riemann surface. These moduli spaces arise in a program to define a new enumerative invariant of 3-manifolds. They are also of independent…

We review a proof of the well know result stating that moduli spaces of stable sheaves with fixed Chern character on a polarized $K3$ surface are deformations of a hyperk\"ahler variety of Type $K3^{[n]}$ (if a suitable numerical hypothesis…

In this paper we study the Brill-Noether theory of sub-line bundles of a general, stable rank-two vector bundle on a curve C with general moduli. We relate this theory to the geometry of unisecant curves on smooth, non-special scrolls with…

We discuss the Picard group of moduli space $\mathcal{K}_g$ of quasi-polarized K3 surfaces of genus $g\leq 12$ and $g\neq 11$. In this range, $\mathcal{K}_g$ is unirational and a general element in $\mathcal{K}_g$ is a complete intersection…

We show that any polarized K3 surface supports special Ulrich bundles of rank 2.