On GIT stability of linear systems of hypersurfaces in projective spaces

In this paper, we consider the problem of classifying linear systems of hypersurfaces (of a fixed degree) in projective space up to projective equivalence. Our main result consists of a complete criterion for (semi)stability in the sense of geometric invariant theory (GIT). As an application, we inspect a few relevant geometric examples recovering, for instance, Miranda’s characterization of GIT stability of pencils of plane cubics. Furthermore, we completely describe GIT stability of Halphen pencils of any index.