On the GIT stability of linear systems of hypersurfaces in projective space

TL;DR

This paper provides a complete GIT stability classification for linear systems of hypersurfaces in projective space, with explicit criteria and applications to specific geometric examples.

Contribution

It introduces an explicit criterion for GIT stability of hypersurface systems and applies it to classify stability of Halphen pencils and other cases.

Findings

Complete GIT stability classification for hypersurface systems

Explicit criterion for stability determination

Application to Halphen pencils and plane cubics

Abstract

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.