A Stratification of the Semistable Locus

Abstract

Consider the following sublocus of $\mathrm{Ch}^{-1}(\mathrm{Chow}_{d}^{\mathit{ss}}) \subset \mathrm{Hilb}_{d}$:

$$\displaystyle{ \mathrm{Ch}^{-1}(\mathrm{Chow}_{ d}^{\mathit{ss}})^{o}:=\{ [X \subset \mathbb{P}^{r}] \in \mathrm{Ch}^{-1}(\mathrm{Chow}_{ d}^{\mathit{ss}}) \subset \mathrm{Hilb}_{ d}\::\: X\text{ is connected}\}. }$$

(10.1)

If d > 2(2g − 2), the condition of being connected is both closed and open in $\mathrm{Ch}^{-1}(\mathrm{Chow}_{d}^{\mathit{ss}})$: it is closed because of its natural interpretation as a topological condition; it is open because $[X \subset \mathbb{P}^{r}] \in \mathrm{Ch}^{-1}(\mathrm{Chow}_{d}^{\mathit{ss}})$ is a reduced curve by the Potential pseudo-stability Theorem 5.1 and therefore X is connected if and only if $h^{0}(X,\mathcal{O}_{X}) = 1$, which is an open condition by upper-semicontinuity.