Related papers: Towards Brill-Noether Theory for Spectral Curves

The divisor theory of the complete graph $K_n$ is in many ways similar to that of a plane curve of degree $n$. We compute the splitting types of all divisors on the complete graph $K_n$. We see that the possible splitting types of divisors…

We compute the rational cohomology groups of the smooth Brill-Noether varieties $G^r_d(C)$, parametrizing linear series of degree $d$ and dimension exactly $r$ on a general curve $C$. As an application, we determine the whole intersection…

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…

We consider coverings of real algebraic curves to real rational algebraic curves. We show the existence of such coverings having prescribed topological degree on the real locus. From those existence results we prove some results on…

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…

This is the expanded text of a series of CIME lectures. We present an algebro-geometric approach to integrable systems, starting with those which can be described in terms of spectral curves. The prototype is Hitchin's system on 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 determine the splitting (isomorphism) type of the normal bundle of a generic genus-0 curve with 1 or 2 components in any projective space, as well as the (sometimes nontrivial) way the bundle deforms locally with a general deformation of…

Given the Prym variety of an \'etale double cover one can define analogues of the classical Brill-Noether loci on Jacobians of curves. Recent work by Lahoz and Naranjo shows that the Brill-Noether locus $V^2$ completely determines the…

In this paper, our aim is to find the relations amongst the cohomology classes of Brill-Noether subvarieties of the moduli space of semistable bundles over an elliptic curve. We obtain results similar to the Poincar\'e relations on a…

A Steiner bundle over the projective 3-space is the kernel in a trivial bundle of a morphism defined by a matrix of linear forms. We produce various Steiner bundles E of rank n such that E(1) has n-1 sections, the dependency locus of which…

In the 1990's, Bertram, Feinberg and Mukai examined Brill-Noether loci for vector bundles of rank 2 with fixed canonical determinant, noting that the dimension was always bigger in this case than the naive expectation. We generalize their…

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…

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…

We prove the injectivity of the Petri map for linear series on a general curve with given ramification at two generic points. We also describe the components of such a set of linear series on a chain of elliptic curves.

In this paper, we study maps from reducible curves $f : C \cup_\Gamma D \to \mathbb{P}^r$. We restrict our attention to two cases: first, when $f|_D$ factors through a hyperplane $H$ and $f|_C$ is transverse to $H$; and second, when $r =…

In this paper we consider the Brill-Noether locus $W_{\underline d}(C)$ of line bundles of multidegree $\underline d$ of total degree $g-1$ having a nonzero section on a nodal reducible curve $C$ of genus $g\geq2$. We give an explicit…

In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least…

The classical Brill-Noether theorem states that a map from a general curve to a projective space deforms in a family of expected dimension as long as its image does not lie in any hyperplane. In this note, we observe, as a direct…

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 $k$…