# Hodge numbers for Quot schemes on curves

Last week Massimo Bagnarol, Barbara Fantechi, Fabio Perroni: On the motive of zero-dimensional Quot schemes on a curve appeared on the arXiv, and they study moduli spaces of sheaves on a curve which I hadn't seen before (at least not for their intrinsic geometry, only as a tool). They describe the class in the Grothendieck ring of $\mathrm{Quot}_C^n(\mathcal{E})$, where $C$ is a smooth projective curve of genus $g$, $\mathcal{E}$ is a vector bundle of rank $r$ on $C$ and $n\geq 1$ is the length of the finite-dimensional quotients of $\mathcal{E}$ which are parametrised.

They show that the class in the Grothendieck ring is independent of $\mathcal{E}$, effectively reducing the computation to an earlier result of Bifet for $\mathcal{E}=\mathcal{O}_C^{\oplus r}$. The formula (see proposition 4.5) is \begin{equation} [\mathrm{Quot}_C^n(\mathcal{E})] = \sum_{\mathbf{n}\in\mathbb{N}^r,|\mathbf{n}|=n} [\operatorname{Sym}{n_1}C]\cdots[\operatorname{Sym}^{n_r}C]\mathbb{L}^{d_{\mathbf{n}}} \end{equation} where $d_{\mathbf{n}}:=\sum_{i=1}^r(i-1)n_i$, so one can then apply the motivic measure $\mathrm{K}_0(\mathrm{Var}/k)\to\mathbb{Z}[x,y]$ given by the Hodge polynomial to get the Hodge diamond. For all of you who wish to use it in Hodge diamond cutter I have good news, as I just added it.

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