# Algebraic Geometry Seminar

#### Burkhardt, Todd, Igusa, Beauville and Rational Quartic Threefolds

**Speaker:**
Ivan Cheltsov, University of Edinburgh

**Location:**
Warren Weaver Hall 201

**Date:**
Tuesday, October 20, 2015, 3:30 p.m.

**Synopsis:**

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. Recently 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.