Equivariant intermediate Jacobians and intersections of two quadrics

TL;DR

This paper provides a concise proof of a theorem relating group actions and geometric properties of intersections of quadrics in projective space, focusing on equivariant Jacobians and invariant lines.

Contribution

It offers a short proof of a theorem connecting G-equivariant geometry of intersections of quadrics with the existence of G-invariant lines.

Findings

A G-equivariant smooth complete intersection of two quadrics is projectively G-linear if and only if it contains a G-invariant line.

The proof simplifies understanding of G-actions on such intersections.

The result links geometric properties with group symmetry in algebraic geometry.

Abstract

We present a short proof of the following theorem of Hassett and Tschinkel: for every finite group $G$, a $G$-equivariant smooth complete intersection of two quadrics in $\mathbb{P}^5_{\mathbb{C}}$ is projectively $G$-linear if and only if it contains a $G$-invariant line.