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. Beauiville proved that every quartic threefold in this pencil is irrational except for Burkhardt and Igusa quartics and possibly two more threefolds. In this talk I will show how to use two constructions of Todd (dated back to 1933 and 1935) to prove that the remaining two quartic threefolds in the pencil are also rational. Then I will present a new proof of this and Beauville's result using birational geometry of Coble fourfold. This is a joint work with Sasha Kuznetsov and Costya Shramov from Moscow. |

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