Coble fourfold, $S_6$-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 $S_6$-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.

57 pages; v2: minor changes; v3: referee's comments taken into account; v4: published version

References in corpus (3)

• Six line configurations and string dualities
• Küchle fivefolds of type c5
• Which quartic double solids are rational?

Cited by in corpus (1)

• Icosahedron in birational geometry