Brill-Noether theory for moduli spaces of sheaves on algebraic varieties

Abstract

Let $X$ be a smooth projective variety of dimension $n$ and let $H$ be an ample line bundle on $X$. Let $M_{X,H}(r;c_1,...,c_s)$ be the moduli space of $H$-stable vector bundles $E$ on $X$ of rank $r$ and Chern classes $c_i(E)=c_i$ for $i=1,...,s:=min\{r,n\}$. We define the Brill-Noether filtration on $M_{X,H}(r;c_1,...,c_s)$ as $W_H^k(r;c_1,...,c_s)=\{E\in M_{X,H}(r;c_1,...,c_s)|h^0(X,E)\ge k\}$ and we realize $W_H^k(r;c_1,...,c_s)$ as the $k$th determinantal variety of a morphism of vector bundles on $M_{X,H}(r;c_1,...,c_s)$, provided $H^i(E)=0$ for $i\ge 2$ and $E\in M_{X,H}(r;c_1,...,c_s)$. We also compute the expected dimension of $W_H^k(r;c_1,...,c_s)$. Very surprisingly we will see that the Brill-Noether stratification allow us to compare moduli spaces of vector bundles on Hirzebruch surfaces stables with respect to different polarizations. We will also study the Brill-Noether loci of the moduli space of instanton bundles and we will see that they have the expected dimension.

19 pages. To appear Forum Math