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Twisted cubics on singular cubic fourfolds—On Starr’s fibration

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Abstract

We show that the Hilbert scheme compactification of the total space of Starr’s fibration on the space of twisted cubics on a cubic hypersurface in \({\mathbb P}^5\) not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight which has a crepant resolution deformation equivalent to the symplectic eightfold constructed from twisted cubics on a smooth cubic fourfold. This yields another proof that the symplectic eightfold and the Hilbert scheme of four points on a K3 surface are deformation equivalent. As a byproduct we obtain similar results for the variety of lines on a singular cubic fourfold.

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Notes

  1. The rational map \(\mathrm{Hilb}^2(S) \dashrightarrow M_1\) is a morphism as soon as Y does not contain a plane.

  2. In Addington’s paper the divisor \({\mathcal C}_6\) of singular cubics is not considered.

  3. Let us denote by \({\widetilde{Y}}\) a desingularization of Y which is rationally connected by the analysis of singularities from [15, Proposition 5.8]. Note that a desingularization of D is dominated by a blowup of \({{\text {Gr}}}(\Omega _{\widetilde{Y}},3)\) which is clearly rationally connected.

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Acknowledgements

It is a great pleasure to thank Jason Starr for explaining me his construction of the Lagrangian fibration, for his generosity in sharing insights, for helpful discussions and for pointing out several questions. Furthermore, I greatly benefited from discussions with Radu Laza and Claire Voisin. The author gratefully acknowledges the support by the DFG through the DFG research Grants Le 3093/1-1, Le 3093/2-1, and Le 3093/3-1.

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  1. Fakultät für Mathematik, Technische Universität Chemnitz, Reichenhainer Straße 39, 09126, Chemnitz, Germany

    Christian Lehn

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  1. Christian Lehn
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Lehn, C. Twisted cubics on singular cubic fourfolds—On Starr’s fibration. Math. Z. 290, 379–388 (2018). https://doi.org/10.1007/s00209-017-2021-x

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