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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?

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Your dimension is correct, this and more can be found in this article.

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