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

Let $X$ be a smooth projective variety of dimension $n$ and let $H$ be an ample line bundle on $X$. Let $M_{X,H}(r;c_1,\mathrm{.}\mathrm{.}\mathrm{.},c_s)$ be the moduli space of $H$-stable vector bundles $E$ on $X$ of rank $r$ and Chern classes $c_i(E)=c_i$ for $i=1,\mathrm{.}\mathrm{.}\mathrm{.},s:=min\{r,n\}$. We define the Brill-Noether filtration on $M_{X,H}(r;c_1,\mathrm{.}\mathrm{.}\mathrm{.},c_s)$ as $W_{H}^{k}(r;c_1,…, c_{s})= {E \in M_{X,H}(r;c_1, …, c_{s}) | h^0(X,E) \geq k }$andwerealize$W_{H}^{k}(r;c_1,…, c_{s})$asthe$k$th determinantal variety of a morphism of vector bundles on $M_{X,H}(r;c_1,\mathrm{.}\mathrm{.}\mathrm{.},c_s)$, provided $H^i(E)=0$ for $i \geq 2$and$E \in M_{X,H}(r;c_1, …, c_{s})$. We also compute the expected dimension of $W_H^k(r;c_1,\mathrm{.}\mathrm{.}\mathrm{.},c_s)$. 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.