Speaker: Ivan Cheltsov (Edinburgh)
Title:Burkhardt, Todd, Igusa, Beauville and rational
quartic threefolds
Abstract: Burkhardt and Igusa quartic threefolds are
classically known to be rational. They generate a pencil of quartics
that all admit an action of the symmetric group of degree six. Bondal
and Prokhorov asked which threefolds in this pencil are rational and
which are not. All these threefolds are singular, so Iskovskikh and
Manin's result cannot be applied here. Moreover, they are all not
factorial, so Mella's result is not applicable either. Recently
Beauville proved irrationality of all but 4 threefolds in this
pencil. Of course, the 4 threefolds skipped by Beauville include
Burkhardt and Igusa quartics. In this talk I will prove that the two
remaining threefolds in this pencil are also rational. The proof is
based on the two constructions of Todd dated back to 1933 and 1935.
This is a joint work with Costya Shramov from Moscow.