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 $W_{ H}^{ k}(r;c_1,..., c_{ s})= \{ E \in M_{ X,H}(r;c_1, ..., c_{ s}) | h^0(X,E) \geq k \}$ 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 $i \geq 2$ and $E \in M_{ X,H}(r;c_1, ..., c_{ s})$. We also compute the expected dimension of $W_{ H}^{ k}(r;c_1,..., c_{ s})$. 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.
Categories: math.AG