  We consider the problem of classifying linear systems of hypersurfaces (of a
fixed degree) in some projective space up to projective equivalence via
geometric invariant theory (GIT). We provide an explicit criterion that solves
the problem completely. As an application, we consider a few relevant geometric
examples recovering, for instance, Miranda's description of the GIT stability
of pencils of plane cubics. Furthermore, we completely describe the GIT
stability of Halphen pencils of any index.
