Abstract
The Debarre-de Jong conjecture predicts that the Fano variety of lines on a smooth Fano hypersurface in \(\mathbb {P}^n\) is always of the expected dimension. We generalize this conjecture to the case of smooth Fano complete intersections and prove that for a smooth Fano complete intersection \(X\subset \mathbb {P}^n\) of hypersurfaces whose degrees sum to at most 7, the Fano variety of lines on X has the expected dimension.
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References
Beheshti, R.: Lines on projective hypersurfaces. J. Reine Angew. Math. 592, 1–21 (2006). https://doi.org/10.1515/CRELLE.2006.020
Beheshti, R.: Hypersurfaces with too many rational curves. Math. Ann. 360(3–4), 753–768 (2014). https://doi.org/10.1007/s00208-014-1024-8
Beheshti, R, Riedl, E.: Linear subspaces of hypersurfaces (2019).arXiv:1903.02481
Collino, A.: Lines on quartic threefolds. J. Lond. Math. Soc. (2) 19(2), 257–267 (1979). https://doi.org/10.1112/jlms/s2-19.2.257. (MR533324)
Debarre, O.: Lines on smooth hypersurfaces Preprint at https://www.math.ens.fr/~debarre/Lines_hypersurfaces.pdf (2003)
Dolgachev, I.V.: Classical Algebraic Geometry. Cambridge University Press, Cambridge (2012)https://doi.org/10.1017/CBO9781139084437, A modern view, MR2964027
Harris, J., Mazur, B., Pandharipande, R.: Hypersurfaces of low degree. Duke Math. J. 95(1) 125–160 (1998) MR1646558
Loughran, D.: (https://mathoverflow.net/users/5101/daniel loughran), Lines on fano complete intersections https://mathoverflow.net/q/147229 (version: 2013-11-07)
Rogora, E.: Varieties with many lines. Manuscripta Math. 82(2), 207–226 (1994). https://doi.org/10.1007/BF02567698MR1256160
Segre, B.: Sulle \(V_n\) contenenti più di \(\infty ^{n-k}S_k\). I and II Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 5, 193–197 (1948) and 275-280 MR36042
Acknowledgements
I would like to thank my advisor Elham Izadi and David Stapleton for the many helpful conversations. In addition, I am very grateful to the anonymous referee for several helpful comments and suggestions, all of which helped improve the exposition. This work was partially supported by NSF grant DMS-1502651.
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Canning, S. On a conjecture on the variety of lines on Fano complete intersections. Annali di Matematica 200, 2127–2131 (2021). https://doi.org/10.1007/s10231-021-01071-z
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DOI: https://doi.org/10.1007/s10231-021-01071-z