Related papers: Brill-Noether loci and strata of differentials

The aim of this note is to exhibit proper first Brill-Noether loci inside the moduli spaces $M_{Y,H}(2;c_1,c_2)$ of $H$-stable rank $2$ vector bundles with fixed Chern classes of a certain type on an Enriques surface $Y$ which is covered by…

We provide a complete classification of Teichm\"uller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichm\"uller curves intersect the boundary of the…

We study Brill-Noether existence on a finite graph using methods from polyhedral geometry and lattices. We start by formulating analogues of the Brill-Noether conjectures (both the existence and non-existence parts) for…

A conjecture of Bondal-Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We…

Generalizing the Martens theorem for line bundles over a curve $C$, we obtain upper bounds on the dimension of the Brill--Noether locus $B^k_{n, d}$ parametrizing stable bundles of rank $n \ge 2$ and degree $d$ over $C$ with at least $k$…

Let $C$ be a smooth projective curve of genus $g>0$. We describe an open locus of Bridgeland stability conditions on the bounded derived category of coherent systems on $C$, and show that stability manifold detects the Brill--Noether theory…

We discuss the role of K3 surfaces in the context of Mercat's conjecture in higher rank Brill-Noether theory. Using liftings of Koszul classes, we show that Mercat's conjecture in rank 2 fails for any number of sections and for any gonality…

Let us consider the locus in the moduli space of curves of genus 2k defined by curves with a pencil of degree k. Since the Brill-Noether number is equal to -2, such a locus has codimension two. Using the method of test surfaces, we compute…

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

Moduli spaces of Abelian and quadratic differentials are stratified by multiplicities of zeroes; connected components of the strata correspond to ergodic components of the Teichmuller geodesic flow. It is known that the strata are not…

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…

The aim of this note is to find upper bounds on the dimension of Brill-Noether locus' inside the moduli space of rank two vector bundles on a smooth algebraic curve. We deduce some consequences of these bounds.

A Laurent polynomial $f$ in two variables naturally describes a projective curve $C(f)$ on a toric surface. We show that if $C(f)$ is a smooth curve of genus at least 7, then $C(f)$ is not Brill-Noether general. To accomplish this, we…

We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the…

Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and…

This paper deals with singularities of genus 2 curves on a general (d_1,d_2)-polarized abelian surface (S,L). In analogy with Chen's results concerning rational curves on K3 surfaces [Ch1,Ch2], it is natural to ask whether all such curves…

In this article we study Brill-Noether loci of moduli space of stable bundles over smooth surfaces. We define Petri map as an analogy with the case of curves. We show the non-emptiness of certain Brill-Noether loci over very general quintic…

In projectivized strata of meromorphic $1$-forms on elliptic curves with only one zero, the locus of residueless differentials is a complex curve endowed with a canonical complex projective structure. Drawing on the multi-scale…

Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component. We announce the general classification theorem: up…

Let $X$ be a non-singular algebraic curve of genus $g$. We prove that the Brill-Noether locus $\bns $ is non-empty if $d= nd' +d'' $ with $0< d'' <2n$, $1\le s\le g$, $d'\geq (s-1)(s+g)/s $, $n\leq d''+(n-k)g$, $(d'',k)\ne(n,n)$. These…