Equivariant rationality of Fano threefolds in the family \textnumero 2.12

TL;DR

This paper investigates when group actions on certain Fano threefolds are linearizable, establishing a criterion based on the Picard group's G-invariant rank, thus advancing understanding of symmetries in algebraic geometry.

Contribution

It provides a necessary and sufficient condition for the linearizability of group actions on specific Fano threefolds, linking it to the Picard group's G-invariant rank.

Findings

Group actions are linearisable iff the G-invariant Picard rank is not 1.

Characterization of symmetries on Fano threefolds of a particular family.

Advances understanding of equivariant rationality in algebraic geometry.

Abstract

We prove that a faithful group action on the smooth complete intersection $X$ of three divisors of bidegree $(1,1)$ in $\p^3\times\p^3$ is linearisable if and only if $\rk(\pic^G(X))\ne1$.