### A characterization of ample vector bundles on a curve.

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Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let ${F}_{X}$ denote the absolute Frobenius morphism of $X$ when the characteristic of $k$ is positive. A vector bundle over $X$ is called virtually globally generated if its pull back, by some finite morphism to $X$ from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of $k$ is positive, a vector bundle $E$ over $X$ is virtually globally generated if and only if ${\left({F}_{X}^{m}\right)}^{*}E\phantom{\rule{0.166667em}{0ex}}\cong \phantom{\rule{0.166667em}{0ex}}{E}_{a}\oplus {E}_{f}$ for...

We prove that a certain Brill-Noether locus over a non-hyperelliptic curve C of genus 4, is isomorphic to the Donagi-Izadi cubic threefold in the case when the pencils of the two trigonal line bundles of C coincide.

Let $G$ be a complex algebraic group, simple and simply connected, $T$ a maximal torus and $W$ the Weyl group. One shows that the coarse moduli space ${M}_{G}\left(X\right)$ parametrizing $S$-equivalence classes of semistable $G$-bundles over an elliptic curve $X$ is isomorphic to $\left[\Gamma \right(T\left){\otimes}_{\mathbf{Z}}X\right]/W$. By a result of Looijenga, this shows that ${M}_{G}\left(X\right)$ is a weighted projective space.

The aim of these notes is to generalize Laumon’s construction [20] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article “ On the geometric Langlands conjecture” by Frenkel, Gaitsgory and Vilonen [11]...

Moduli spaces of vector bundles on families of non-singular curves are usually compactified by considering (slope)semistable bundles on stable curves. Alternatively, one could consider Hilbert-stable curves in Grassmannians. We study some properties of the latter and compare them with similar properties of curves coming from the former compactification. This leads to a new interpretation of the moduli space of (semi)stable torsion-free sheaves on a fixed nodal curve. One can present it as a quotient...

We prove that the global geometric theta-lifting functor for the dual pair $(H,G)$ is compatible with the Whittaker functors, where $(H,G)$ is one of the pairs $({\mathrm{S}\mathbb{O}}_{2n},{\mathbb{S}p}_{2n})$, $({\mathbb{S}p}_{2n},{\mathrm{S}\mathbb{O}}_{2n+2})$ or $({\mathbb{G}\mathrm{L}}_{n},{\mathbb{G}\mathrm{L}}_{n+1})$. That is, the composition of the theta-lifting functor from $H$ to $G$ with the Whittaker functor for $G$ is isomorphic to the Whittaker functor for $H$.

We give a characterization of conformal blocks in terms of the singular cohomology of suitable smooth projective varieties, in genus $0$ for classical Lie algebras and ${G}_{2}$.