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 𝔖6-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 𝔖6-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 𝔖6-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
Milestones
Received: 2 February 2019
Revised: 1 July 2019
Accepted: 1 September 2019
Published: 15 March 2020
Authors
Ivan Cheltsov
Department of Mathematics
University of Edinburgh
United Kingdom
National Research University Higher School of Economics
Russian Federation
Alexander Kuznetsov
Algebraic Geometry Section
Steklov Mathematical Institute of Russian Academy of Sciences
Moscow
Russia
National Research University Higher School of Economics
Russian Federation
Konstantin Shramov
Algebraic Geometry Section
Steklov Mathematical Institute of Russian Academy of Sciences
Moscow
Russia
National Research University Higher School of Economics
Russian Federation