Stability of nets of quadrics in $\mathbb{P}^5$ and associated discriminants

TL;DR

This paper studies the geometric invariant theory (GIT) stability of nets of quadrics in projective 5-space, exploring the relationship between the stability of the associated discriminant sextic curve and the properties of the surface defined by the net.

Contribution

It establishes conditions under which the stability of the discriminant implies the stability of the net, and relates singularities of the surface to those of the discriminant curve.

Findings

If the surface is normal and the discriminant is stable, then the net is stable.

If the discriminant is stable and the surface has reduced discriminant, then the net is stable.

Surfaces with simple singularities have discriminants with simple singularities.

Abstract

Let $S$ be a complete intersection surface defined by a net $\Lambda$ of quadrics in $\mathbb P^5$. In this paper we analyze GIT stability of nets of quadrics in $\mathbb P^5$ up to projective equivalence, and discuss some connections between a net of quadrics and the associated discriminant sextic curve. In particular, we prove that if $S$ is normal and the discriminant $\Delta(S)$ of $S$ is stable then $\Lambda$ is stable. And we prove that if $S$ has the reduced discriminant and $\Delta(S)$ is stable then $\Lambda$ is stable. Moreover, we prove that if $S$ has simple singularities then $\Delta(S)$ has simple singularities.