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 $0 \le β< γ$ be two integers, ($γ\ge 2)$, such that $2γ-3β>0$; then ${\cal M}_{I\!\! P^3}(0,2γ^2-3β^2)$ is singular (the case $β=0$ was previously proved by M. Maggesi). 2) Let $0 \le β< γ$ be two odd integers ($γ\ge 5)$, such that $2γ-3β+1>0$; then ${\cal M}_{I\!\! P^3}(-1,2(γ/2)^2-3(β/2)^2+1/4)$ is singular. In particular ${\cal M}_{I\!\! P^3}(0,5)$, ${\cal M}_{I\!\! P^3}(-1,6)$ are singular.