Maximum number of points on intersection of a cubic surface and a non-degenerate Hermitian surface

Peter Beelen, Mrinmoy Datta

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Abstract

In 1991 Sørensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree d and a non-degenerate Hermitian surface in P3(Fq2). The conjecture was proven to be true by Edoukou in the case when d = 2. In this paper, we prove that the conjecture is true for d = 3. For q ≥ 4, we also determine the second highest number of rational points on the intersection of a cubic surface and a non-degenerate Hermitian surface. Finally, we classify all the cubic surfaces that admit the highest and, for q ≥ 4, the second highest number of points in common with a non-degenerate Hermitian surface. This classification disproves a conjecture proposed by Edoukou, Ling and Xing.
Original languageEnglish
JournalMoscow Mathematical Journal
Volume20
Issue number3
Pages (from-to)453-474
ISSN1609-3321
DOIs
Publication statusPublished - 2020

Keywords

  • Hermitian surfaces
  • Cubic surfaces
  • Intersection of surfaces
  • Rational points

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