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Let $X$ be a smooth projective variety of dimension $n$ and let $H$ be anample line bundle on $X$. Let $M_{X,H}(r;c_1, ..., c_{s})$ be the moduli spaceof $H$-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-Noetherfiltration 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 morphismof 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 expecteddimension of $W_{H}^{k}(r;c_1,..., c_{s})$. Very surprisingly we will see thatthe Brill-Noether stratification allow us to compare moduli spaces of vectorbundles on Hirzebruch surfaces stables with respect to different polarizations.We will also study the Brill-Noether loci of the moduli space of instantonbundles and we will see that they have the expected dimension.

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