Let $S$ be a projective smooth surface over $\mathbb{C}$ and let $E$ be a rank-$r$ vector bundle on $S$. Having fixed an Hilbert polynomial $P$, which is the expected dimension of the Quot-scheme $Quot\_S(E,P)$?



Expected dimension of Quot-schemes on algebraic surface
Let $F$ be a quotient of $E$ with Hilbert polinomial $P$ and $[F]$ the corresponding point of $Quot\_S(E,P)$. Let $K = Ker(E \to F)$. Then the tangent space to $Quot\_S(E,P)$ at $[F]$ is $Hom(K,F)$ and the obstruction space is $Ext^1(E,F)$. So, the expected dimension is 
$$
\chi(K,F) = \sum (-1)^i \dim Ext^i(K,F),
$$
which can be computed in terms of $P$ and the Chern classes of $S$ by Riemann--Roch.


