Hypersurfaces with defect

Lindner, Niels

doi:10.1016/j.jalgebra.2020.02.022

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Cited by 2 publications

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“…The energetic reader will have no trouble extending most of our results to universal families of hypersurfaces; or reformulating most of them in terms of cohomological "defect" (in the sense of e.g. [Lin20]), via Proposition 1.7 below; or both. But we will focus on the V c 's and their point counts, since they play-in a fashion building on [Hoo86]-a central role in the aforementioned work of the author.…”

Section: Introductionmentioning

confidence: 99%

Dichotomous point counts over finite fields

Wang¹

2022

Preprint

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We establish a near dichotomy between randomness and structure for the point counts of arbitrary projective cubic threefolds over finite fields. Certain "special" subvarieties, not unlike those in the Manin conjectures, play a distinguished role. This suggests that similar (near) dichotomies may exist more generally for other natural families of varieties.We also establish partial results for projective hypersurfaces in general, using tools including "worst-case" cohomological results of Skorobogatov or Katz, "average-case" results of Lindner, the existence theory of Lefschetz pencils, and the theory of discriminants. This continues a line of inquiry initiated by Hooley.

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Section: Introductionmentioning

confidence: 99%