Let $\mathcal{H}$ be a coherent sheaf on a projective variety $X$. We say that a sheaf $\mathcal{E}$ is of $\textit{pure dimension}$ if for all non-trivial coherent subsheaves $\mathcal{E'} \subset \mathcal{E}$ we have $\dim(\mathcal{E'}) = \dim(\mathcal{E})$. It can be shown that the subset of points of the $Quot$ scheme of $\mathcal{H}$ corresponding to pure sheaves is open. Consider now a point $\rho : \mathcal{H} \rightarrow \mathcal{F}$ in the closure of that open subset. I would like to understand why in this case, the point $\rho$ can be deformed into one that corresponds to a pure sheaf.

In the book of Huybrechts & Lehn - The geometry of moduli spaces of sheaves, the authors say that this condition on $\rho$ means that there exists a smooth connected curve $C$ and a flat family of sheaves $\mathcal{E}$ parametrized by $C$ such that $\mathcal{E}_0 \cong \mathcal{F}$ for some closed point $0 \in C$ and such that $\mathcal{E}_s$ is pure for all $s \in C - {0}$. Why does a curve like this always exist in the case above i.e. when $\mathcal{F}$ is in the closure of the open set of pure sheaves?

  • $\begingroup$ This has nothing to do with pure sheaves. If $Z$ is a variety, $U$ open and $p$ in the closure of $U$, then either $p\in U$ or there exists an irreducible curve $C\subset Z$ with $C\cap U\neq\emptyset$ and $p\in C$. This is easy to prove by analyzing what happens in a neighborhood of $p$. $\endgroup$ – Mohan May 18 '16 at 16:57

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