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

Abstract

Let ${\mathcal{M}}_{I{P}^3}(c_1,c_2)$ 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 ${\mathcal{M}}_{I{P}^3}(0,2{\gamma }^2-3{\beta }^2)$ 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 ${\mathcal{M}}_{I{P}^3}(-1,2(\gamma /2{)}^2-3(\beta /2{)}^2+1/4)$ is singular. In particular ${\mathcal{M}}_{I{P}^3}(0,5)$, ${\mathcal{M}}_{I{P}^3}(-1,6)$ are singular.

7 pages, Plain Tex