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

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 ${\mathfrak{𝔖}}_6$-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 ${\mathfrak{𝔖}}_6$-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 ${\mathfrak{𝔖}}_6$-representations.