Algebra & Number Theory

Coble fourfold, $\mathfrak S_6$-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.

Article information

Source
Algebra Number Theory, Volume 14, Number 1 (2020), 213-274.

Dates
Received: 2 February 2019
Revised: 1 July 2019
Accepted: 1 September 2019
First available in Project Euclid: 7 April 2020

Permanent link to this document
https://projecteuclid.org/euclid.ant/1586224824

Digital Object Identifier
doi:10.2140/ant.2020.14.213

Mathematical Reviews number (MathSciNet)
MR4076812

Zentralblatt MATH identifier
07180786

Citation

Cheltsov, Ivan; Kuznetsov, Alexander; Shramov, Konstantin. Coble fourfold, $\mathfrak S_6$-invariant quartic threefolds, and Wiman–Edge sextics. Algebra Number Theory 14 (2020), no. 1, 213--274. doi:10.2140/ant.2020.14.213. https://projecteuclid.org/euclid.ant/1586224824

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