$\mathfrak A_5$-equivariant geometry of quadric threefolds

TL;DR

This paper classifies certain symmetric geometric structures on smooth quadric threefolds under the action of the alternating group, revealing their non-linearizable symmetries and solidness.

Contribution

It provides a classification of $rak A_5$-equivariant Mori fiber spaces birational to smooth quadric threefolds with fixed-point free actions.

Findings

Quadric threefolds are $G$-solid.

The $G$-actions are not projectively linearizable.

Classification of $G$-Mori fiber spaces related to quadric threefolds.

Abstract

We classify $G$-Mori fibre spaces equivariantly birational to smooth quadric threefolds with fixed-point free actions of the alternating group $G=\mathfrak A_5$. We deduce that such quadric threefolds are $G$-solid and the $G$-actions on them are not projectively linearizable.