Twisted cubics on cubic fourfolds

Christian Lehn; Manfred Lehn; Christoph Sorger; Duco van Straten

doi:10.1515/crelle-2014-0144

Article

Twisted cubics on cubic fourfolds

Christian Lehn , Manfred Lehn , Christoph Sorger and Duco van Straten

Published/Copyright: April 28, 2015

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2017 Issue 731

Abstract

We construct a new twenty-dimensional family of projective eight-dimensional irreducible holomorphic symplectic manifolds: the compactified moduli space M3⁢(Y) of twisted cubics on a smooth cubic fourfold Y that does not contain a plane is shown to be smooth and to admit a contraction M3⁢(Y)→Z⁢(Y) to a projective eight-dimensional symplectic manifold Z⁢(Y). The construction is based on results on linear determinantal representations of singular cubic surfaces.

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: Le 3093/1-1; SFB Transregio 45 Bonn-Mainz-Essen

Funding statement: The first-named author was supported by the ANR program VHSMOD, Grenoble, the Labex Irmia, Strasbourg, and, during the revision of the article, by the DFG through the research grant Le 3093/1-1. The third-named author would like to thank the SFB Transregio 45 Bonn-Mainz-Essen and the Max-Planck-Institut für Mathematik Bonn for their hospitality.

Acknowledgements

This project got launched when L. Manivel pointed out to one of us that the natural morphism M3⁢(Y)→Grass⁡(6,4) to the Grassmannian admits a Stein factorisation M3⁢(Y)→ZStein→Grass⁡(6,4) such that ZStein→Grass⁡(6,4) has degree 72. We are very grateful to him for sharing this idea with us. We have profited from discussions with C. von Bothmer, I. Dolgachev, E. Looijenga and L. Manivel.

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Received: 2013-10-9

Revised: 2014-9-15

Published Online: 2015-4-28

Published in Print: 2017-10-1

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https://doi.org/10.1515/crelle-2014-0144