On the Quot scheme $\mathrm{Quot}^{l}(\mathscr{E})$

Samuel Stark

Published 2021-07-08Version 1

We study the Quot scheme $\mathrm{Quot}^l(\mathscr{E})$ of length $l$ coherent sheaf quotients of a locally free sheaf $\mathscr{E}$ on a smooth projective surface. We conjecture that the singularities of $\mathrm{Quot}^l(\mathscr{E})$ are rational. For $l=2$ we use the ADHM description to show that this conjecture holds, and in fact exhibit a modular resolution of singularities; this leads us to a sheaf-theoretic generalisation of Severi's double point formula. Given a short exact sequence $0\rightarrow\mathscr{E}^{'}\rightarrow\mathscr{E}\rightarrow\mathscr{E}^{''}\rightarrow 0$, we show that $\mathrm{Quot}^l(\mathscr{E}^{''})\subset\mathrm{Quot}^l(\mathscr{E})$ is the zero scheme of a regular section of the tautological sheaf $\mathscr{E}^{'\vee[l]}$, which results in a relation between the (ordinary and virtual) fundamental classes of $\mathrm{Quot}^l(\mathscr{E}^{''})$ and $\mathrm{Quot}^l(\mathscr{E})$.