Given a locally free sheaf $E$ of rank $r$ on a (smooth, projective, algebraic) surface, I want to know the dimension of the scheme parametrizing the zero-dimensional (meaning they have zero dimensional support) quotients of $E$ of length $k$.


I believe the answer should be $(r+1)k$. Indeed, the generic thing seems to be a quotient of the form $\bigoplus\_{i=1}^k \mathbb C\_{p\_i}$, with an $r-1$ choice of dimensions for a map $E\_{p\_i} \rightarrow \mathbb C\_{p\_i}$ for each $i$, and $2k$ dimensions of choices for the $p\_i$.


**Question**
Is this the correct dimension? If so, does anyone have a reference with a rigorous argument?



Dimension of Quot scheme of zero dimensional quotients of a locally free sheaf
Your dimension is correct, this and more can be found in this [article.](http://arxiv.org/abs/alg-geom/9704016)


