Stability of hypersurface sections of quadric threefolds

TL;DR

This paper investigates the geometric invariant theory (GIT) stability of hypersurface sections of quadric threefolds in projective 4-space, establishing stability conditions based on degree and singularity types.

Contribution

It provides new criteria for GIT stability and semistability of hypersurface sections of quadric threefolds depending on degree and singularities.

Findings

For degree d ≥ 4, semi-log canonical singularities imply G-stability.

For degree d ≥ 3, semi-log canonical singularities imply G-semistability.

Abstract

Let $S$ be a complete intersection of a smooth quadric 3-fold $Q$ and a hypersurface of degree $d$ in ${\mathbb P}^4$. In this paper we analyze GIT stability of $S$ with respect to the natural $G=SO(5, {\mathbb C})$-action. We prove that if $d\ge 4$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-stable. Also, we prove that if $d\ge 3$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-semistable.