Related papers: A refined Brill-Noether theory over Hurwitz spaces

We study closures of GL_2(R)-orbits on the total space of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that, in the generic stratum, such manifolds are the whole stratum,…

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…

Let C be a smooth projective curve over the field of the complex numbers. We consider Brill-Noether loci over the moduli of maps from C to the Grassmannian G(m,n) and the corresponding Quot schemes of quotients of a trivial vector bundle on…

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 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 $X$ be a smooth projective curve of genus $g$ over the field $\mathbb{C}$. Let $M_{X}(2,L)$ denote the moduli space of stable rank $2$ vector bundles on $X$ with fixed determinant $L$ of degree $2g-1$. Consider the Brill-Noether…

For a projective variety X, a line bundle L on X and r a natural number we consider the r-th Brill-Noether locus W^r(L,X):={\eta\in Pic^0(X)|h^0(L+\eta)\geq r+1}: we describe its natural scheme structure and compute the Zariski tangent…

A Brill-Noether locus is a subscheme of the moduli of bundles E over a curve C defined by requiring E to have a given number of sections, or homomorphisms from another bundle. There are a number of different types, that can be treated by…

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…

Let $G$ be a finite graph of genus $g$. Let $d$ and $r$ be non-negative integers such that the Brill-Noether number is non-negative. It is known that for some $k$ sufficiently large, the $k$-th homothetic refinement $G^{(k)}$ of $G$ admits…

We show that the canonical bundle of the Hurwitz stack classifying covers of genus g>1 and degree k>2 of the projective line is big. We show that all coarse moduli spaces of trigonal curves of genus g>1 are of general type.

Let $C$ be a smooth projective complex curve of genus $g \geq 2$. We investigate the Brill-Noether locus consisting of stable bundles of rank 2 and determinant $L$ of odd degree $d$ having at least $k$ independent sections. This locus…

We explain a strategy for distinguishing Brill-Noether loci in the moduli space of curves by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill-Noether loci…

In this paper, we survey recent developments in the Brill-Noether Theory of higher rank vector bundles on complex projective surfaces. We focus on weak Brill-Noether Theorems on rational and K-trivial surfaces and their applications.

In this short note, I point out that results of Ballico and Kool--Shende--Thomas together imply that on $K3$, Enriques, and Abelian surfaces, if $L$ is a very ample and $(2p_a(L)-2g-1)$-spanned line bundle, then the equigeneric Severi…

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…

The Brill-Noether Theorem gives necessary and sufficient conditions for the existence of a linear series. Here we consider a general n-fold, etale cyclic cover p of a curve C of genus g and investigate for which numbers r,d a linear series…

Our purpose in this paper is to construct new examples of twisted Brill Noether loci on curves of genus g greater than 2 with negative expected dimension. We begin by completing the proof of Butler's conjecture for coherent systems of…

Let $E$ be a vector bundle of rank $r\geq 2$ on a smooth projective curve $C$ of genus $g \geq 2$ over an algebraically closed field $K$ of arbitrary characteristic. For any integer with $1\le k\le r-1$ we define $${\se}_k(E):=k\deg…

The geometry of Sp(3)/U(3) as a subvariety of Gr(3,6) is explored to explain several examples given by Mukai of non-abelian Brill-Noether loci, and to give some new examples. These examples identify Brill-Noether loci of vector bundles on…