On singularities of ${\cal M}_{I\!\! P^3}(c_1,c_2)$

Abstract

Let ${\cal M}_{I\!\! P^3}(c_1,c_2)$ be the moduli space of stable rank-$2$ vector bundles on $I\!\! P^3$ with Chern classes $c_1$, $ c_2$. We prove the following results. 1) Let $0 \le \beta < \gamma $ be two integers, ($\gamma \ge 2)$, such that $2\gamma-3\beta>0$; then ${\cal M}_{I\!\! P^3}(0,2\gamma^2-3\beta^2)$ is singular (the case $\beta =0$ was previously proved by M. Maggesi). 2) Let $0 \le \beta < \gamma $ be two odd integers ($\gamma \ge 5)$, such that $2\gamma-3\beta+1>0$; then ${\cal M}_{I\!\! P^3}(-1,2(\gamma/2)^2-3(\beta/2)^2+1/4)$ is singular. In particular ${\cal M}_{I\!\! P^3}(0,5)$, ${\cal M}_{I\!\! P^3}(-1,6)$ are singular.