Group action on instanton bundles over $\PP^3$

Abstract

Denote by MI(k) the moduli space of k-instanton bundles E of rank 2 on $\PP^3=\PP(V)$ and by $Z_k(E)$ the scheme of k-jumping lines. We prove that $[E]\in MI(k)$ is not stable for the action of SL(V) if $Z_k(E)\neq\emptyset$. Moreover $\dim Sym(E)\ge 1$ if $length Z_k(E)\ge 2$. We prove also that E is special if and only if $Z_k(E)$ is a smooth conic. The action of SL(V) on the moduli of special instanton bundles is studied in detail.