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.