Seminar Algebra and Geometry: Ivan Cheltsov (Edinburgh)

Igusa quartic and and Wiman-Edge sextics

The automorphism group of Igusa quartic is the symmetric group of degree 6. There are other quartic threefolds that admits a faithful action of this group. One of them is the famous Burkhardt quartic threefold. Together they form a pencil that contains all $\mathbb{S}_6$-symmetric quartic threefolds.
Arnaud Beauville proved that all but four of these quartic threeffolds are irrational. Later Cheltsov and Shramov proved that the remaining threefolds in this pencil are rational. In this talk, I will give an alternative prove of both these 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 $\mathbb{S}_6$-symmetric quartic threefolds are birational to conic bundles over quintic del Pezzo surfaces whose degeneration curves are contained in the pencil studied by Wiman and Edge.
This is a joint work with Alexander Kuznetsov and Constantin Shramov (Moscow).

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