Algebra & Number Theory

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

Ivan Cheltsov, Alexander Kuznetsov, and Konstantin Shramov

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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 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

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

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

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Zentralblatt MATH identifier

Primary: 14E08: Rationality questions [See also 14M20]
Secondary: 14E05: Rational and birational maps 14J30: $3$-folds [See also 32Q25] 14J35: $4$-folds 14J45: Fano varieties

Fano varieties Igusa quartic conic bundle del Pezzo surface Wiman–Edge pencil


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.

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