Related papers: The hyperelliptic theta map and osculating project…

Let $C$ be a hyperelliptic curve of genus $g \geq 3$. We give a new description of the theta map for moduli spaces of rank 2 semistable vector bundles with trivial determinant. In orther to do this, we describe a fibration of (a birational…

In this paper we investigate generalized theta divisors $\Theta_r$ in the moduli spaces $\mathcal{U}_C(r,r)$ of semistable vector bundles on a curve $C$ of genus $2$. We provide a desingularization $\Phi$ of $\Theta_r$ in terms of a…

Over a smooth complex projective curve $C$ of genus $g$ let $\M (n,d)$ be the moduli space of semistable bundles of rank $n$ and degree $d$ on $C$, and $\SM (n,L)$, the moduli space of those bundles whose determinant is isomorphic to a…

Let $X$ be the moduli space of semistable rank 2 vector bundles over a smooth curve C of genus $g \ge 2$ and $\theta : X \to PH^0(L)^*$ be the map associated to the generalized theta divisor L on X. We prove that for C not hyperelliptic,…

Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed…

We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on $Pic^{g-1}C$ which are linearly equivalent to $2\Theta$. The embedded tangent space at a…

Let U(r) be the moduli space of rank r vector bundles with trivial determinant on a smooth curve of genus 2. The map theta_r: U(r) -> |r Theta|, which associates to a general bundle its theta divisor, is generically finite. In this paper we…

In this paper we study $G$-Higgs bundles over an elliptic curve when the structure group $G$ is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable…

Let C be a curve of genus g, and let SU(r) be the moduli space of vector bundles of rank r on C, with trivial determinant. A general E in SU(r) defines a theta divisor in the linear system |r Theta|, where Theta is the canonical theta…

Let $C$ be an algebraic smooth complex curve of genus $g>1$. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on $C$ and the comparison of different type of…

In a 1993 article, G. Faltings gave a new construction of the moduli space $U$ of semistable vector bundles on a smooth curve $X$, avoiding geometric invariant theory. Roughly speaking, Faltings showed that the normalisation $B$ of the ring…

The level moduli space $A_g^{4,8}$ is mapped to the projective space by means of gradients of odd Theta functions, such a map turning out no to be injective in the genus 2 case. In this work a congruence subgroup $\Gamma$ is located between…

Let $C$ be an algebraic smooth complex genus $g>1$ curve. The object of this paper is the study of the birational structure of the coarse moduli space $U_C(r,0)$ of semi-stable rank r vector bundles on $C$ with degree 0 determinant and of…

We discuss the projectivity of the moduli space of semistable vector bundles on a curve of genus $g\geq 2$. This is a classical result from the 1960s, obtained using geometric invariant theory. We outline a modern approach that combines the…

Let $\text{M}_C( 2, \mathcal{O}_C) \cong \mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let $\beta_C$ denote…

Let $C$ be a curve of genus two. We denote by $SU_C(3)$ the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over $C$, and by $J^d$ the variety of line bundles of degree $d$ on $C$. In particular, $J^1$ has a…

This is a (short) survey lecture on the "theta map" from the moduli space of SL_r bundles on a curve C to the projective space of r-th order theta functions on JC . Some recent results and a few open problems about that map are discussed.

In this paper we introduce the elementary notion of Pl\"ucker form of a pair $(E,S)$, where $E$ is a vector bundle of rank $r$ on a smooth, irreducible, complex projective variety $X$ and $S \subset H^0(E)$ is a subspace of dimension $rm$.…

Let $C$ be a smooth complex irreducible projective curve of genus $g$ with general moduli, and let $(L,H^0(L))$ be a generated complete linear series of type $(d,r+1)$ over $C$. The syzygy bundle, denoted by $M_L$, is the kernel of the…

Given a vector bundle $E$ of rank $r$ and degree $d$ on a curve $C$ of genus $g$, one can associate to $E$ in a natural way several other vector bundles. For example, one can take wedge powers of $E$. If $E$ is generated by global sections,…