Coble fourfold, 𝔖6-invariant quartic threefolds, and Wiman–Edge sextics

Cheltsov, Ivan; Kuznetsov, Alexander; Shramov, Konstantin

doi:10.2140/ant.2020.14.213

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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 $𝔖_{6}$ -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.

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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

Vol. 14, No. 1, 2020