Abstract

The automorphism group of Igusa quartic is the symmetric group of degree 6. There are other quartic threefolds that admit a faithful action of this group. One of them is the famous Burkhardt quartic threefold. Together they form a pencil that contains all $\mathfrak{S}_6$-symmetric quartic threefolds. Arnaud Beauville proved that all but four of them are irrational, while Burkhardt and Igusa quartic are known to be rational. Cheltsov and Shramov proved that the remaining two threefolds in this pencil are also rational. In this talk, I will give an alternative prove of both these (irrationality and rationality) results. To do this, I will describe Q-factorizations of the double cover of the four-dimensional projective space branched over the Igusa quartic, which is known as Coble fourfold. Using this, I will show that $\mathfrak{S}_6$-symmetric quartic threefolds are birational to conic bundles over the quintic del Pezzo surface whose degeneration curves are contained in the pencil studied by Wiman and Edge. This is a joint work with Sasha Kuznetsov and Costya Shramov from Moscow.