Burkhardt quartic, Barth sextic, and the icosahedron

TL;DR

This paper investigates two rational Fano threefolds with icosahedral symmetry, proving their birational superrigidity and providing new embeddings of the group into the Cremona group.

Contribution

It establishes the $rak{A}_5$-birational superrigidity of the Burkhardt quartic and Barth sextic double cover, introducing new group embeddings into the Cremona group.

Findings

Both threefolds are $rak{A}_5$-Fano varieties.

They are $rak{A}_5$-birationally superrigid.

New embeddings of $rak{A}_5$ into the Cremona group are obtained.

Abstract

We study two rational Fano threefolds with an action of the icosahedral group $\mathfrak{A}_5$. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are $\mathfrak{A}_5$-Fano varieties that are $\mathfrak{A}_5$-birationally superrigid. This gives two new embeddings of the group $\mathfrak{A}_5$ into the space Cremona group.