Coble fourfold, $S_6$-invariant quartic threefolds, and Wiman-Edge sextics

TL;DR

This paper studies the geometry of $S_6$-invariant quartic threefolds, showing they are birational to conic bundles over a del Pezzo surface, and explores their rationality and divisor class groups.

Contribution

It constructs resolutions of the Coble fourfold and establishes a birational classification of $S_6$-invariant quartics as conic bundles, providing new insights into their rationality.

Findings

$S_6$-invariant quartics are birational to conic bundles over the quintic del Pezzo surface.

Some quartics are proven to be rational, others irrational.

The Weil divisor class groups are described as $S_6$-representations.

Abstract

We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all $S_6$-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman-Edge pencil. As an application, we check that $S_6$-invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as $S_6$-representations.