Gushel-Mukai varieties with many symmetries and an explicit irrational Gushel-Mukai threefold

TL;DR

This paper constructs explicit Gushel-Mukai threefolds with large automorphism groups and proves their non-rationality using intermediate Jacobian symmetries, advancing understanding of symmetries in Fano varieties.

Contribution

It provides explicit examples of Gushel-Mukai threefolds with large automorphism groups and demonstrates their non-rationality through intermediate Jacobian analysis.

Findings

Explicit construction of a smooth Fano threefold with a faithful PSL(2,F11) action.

Proof of non-rationality of the constructed threefold.

Construction of Gushel-Mukai varieties with large automorphism groups.

Abstract

We construct an explicit complex smooth Fano threefold with Picard number 1, index 1, and degree 10 (also known as a Gushel-Mukai threefold) and prove that it is not rational by showing that its intermediate Jacobian has a faithful $\mathrm{PSL}(2,\mathbf{F}_{11}) $-action. Along the way, we construct Gushel-Mukai varieties of various dimensions with rather large (finite) automorphism groups. The starting point of all these constructions is an Eisenbud-Popescu-Walter sextic with a faithful $\mathrm{PSL}(2,\mathbf{F}_{11}) $-action discovered by the second author in 2013.