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Hypersurfaces with too many rational curves

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Abstract

We study smooth hypersurfaces of degree \(d\ge n+1\) in \(\mathbf{P}^n\) whose spaces of smooth rational curves of low degrees are larger than expected, and show that under certain conditions, the primitive part of the middle cohomology of such hypersurfaces have non-trivial Hodge substructures. As an application, we prove that the space of lines on any smooth Fano hypersurface of degree \(d \le 8\) in \(\mathbf{P}^n\) has the expected dimension \(2n-d-3\).

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Acknowledgments

The author would like to thank Ravindra Girivaru and Matt Kerr for many helpful discussions.

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  1. Department of Mathematics, Washington University in St. Louis, St. Louis, MO, 63130, USA

    Roya Beheshti

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  1. Roya Beheshti
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Correspondence to Roya Beheshti.

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Beheshti, R. Hypersurfaces with too many rational curves. Math. Ann. 360, 753–768 (2014). https://doi.org/10.1007/s00208-014-1024-8

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  • DOI: https://doi.org/10.1007/s00208-014-1024-8

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