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Coble fourfold,
$\mathfrak S_6$-invariant quartic threefolds, and Wiman–Edge
sextics
Ivan Cheltsov, Alexander Kuznetsov and Konstantin Shramov |
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Vol. 14 (2020), No. 1, 213–274
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DOI: 10.2140/ant.2020.14.213
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Abstract |
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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 -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 -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 -representations. |
Keywords
Fano varieties, Igusa quartic, conic bundle, del Pezzo
surface, Wiman–Edge pencil
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Mathematical Subject Classification 2010
Primary: 14E08
Secondary: 14E05, 14J30, 14J35, 14J45
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Milestones
Received: 2 February 2019
Revised: 1 July 2019
Accepted: 1 September 2019
Published: 15 March 2020
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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 |
