Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let M r,Lss denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(M r,Lss) = ℤ, identify the ample generator, and deduce that M r,Lss is locally factorial. In characteristic zero, this has already been proved by Drézet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant...

Sia $X$ una curva liscia di genere $g\ge 2$ ed $A$, $B$ fasci coerenti su $X$. Sia ${\mu }_{A,B}:H^0\left(X,A\right)\otimes H^0\left(X,B\right)\to H^0\left(X,A\otimes B\right)$ l'applicazione di moltiplicazione. Qui si dimostra che ${\mu }_{A,B}$ ha rango massimo se $A\cong {\omega }_X$ e $B$ è un fibrato stabile generico su $X$. Diamo un'interpretazione geometrica dell'eventuale non-surgettività di ${\mu }_{A,B}$ quando $A,B$ sono fibrati in rette generati da sezioni globali e $\mathrm{deg}\left(A\right)+\mathrm{deg}\left(B\right)\ge 3g-1$. Studiamo anche il caso $\text{dim}\left(\text{Coker}\left({\mu }_{A,B}\right)\right)\ge 2$.

Let C be an elliptic curve and E, F polystable vector bundles on C such that no two among the indecomposable factors of E + F are isomorphic. Here we give a complete classification of such pairs (E,F) such that E is a subbundle of F.

Let X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGLr (ℂ)-bundles over X; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of N is trivial.

In this article, we develop a geometric method to construct solutions of the classical Yang–Baxter equation, attaching a family of classical $r$-matrices to the Weierstrass family of plane cubic curves and a pair of coprime positive integers. It turns out that all elliptic $r$-matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin’s classication and prove...

The theory of Whittaker functors for $G{L}_n$ is an essential technical tools in Gaitsgory’s proof of the Vanishing Conjecture appearing in the geometric Langlands correspondence. We define Whittaker functors for $G𝕊p_4$ and study their properties. These functors correspond to the maximal parabolic subgroup of $G𝕊p_4$, whose unipotent radical is not commutative.We also study similar functors corresponding to the Siegel parabolic subgroup of $G𝕊p_4$, they are related with Bessel models for $G𝕊p_4$ and Waldspurger models for $G{L}_2$.We...