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On a conjecture on the variety of lines on Fano complete intersections


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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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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Correspondence to Samir Canning.

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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).

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  • Fano variety of lines
  • Complete intersections
  • Debarre-de Jong conjecture

Mathematics Subject Classification

  • 14J45