Group Actions on Vector Bundles on the Projective Plane

We study the action of the group of automorphisms of the projective plane on the Maruyama scheme of sheaves ${\mathcal{M}}_{P^2}(r,c_{1,}{c}_2)$ of rank $r$ and Chern classes $c_1=0$ and $c_2=n$ and obtain sufficient conditions for unstability in the sense of Mumford’s geometric invariant theory. The conditions are in terms of the splitting behaviour of sheaves when restricted to lines in the projective plane. A very strong parallel is observed with Mumford’s theory for the action of the automorphism group of the projective plane on the spaces of curves of a fixed degree.