#### Vol. 14, No. 1, 2020

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

### Ivan Cheltsov, Alexander Kuznetsov and Konstantin Shramov

Vol. 14 (2020), No. 1, 213–274
DOI: 10.2140/ant.2020.14.213
##### 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.

##### Keywords
Fano varieties, Igusa quartic, conic bundle, del Pezzo surface, Wiman–Edge pencil
##### Mathematical Subject Classification 2010
Primary: 14E08
Secondary: 14E05, 14J30, 14J35, 14J45