## 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 P

^{3}(F*). 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*_{q2}*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 language | English |
---|---|

Journal | Moscow Mathematical Journal |

Volume | 20 |

Issue number | 3 |

Pages (from-to) | 453-474 |

ISSN | 1609-3321 |

DOIs | |

Publication status | Published - 2020 |

## Keywords

- Hermitian surfaces
- Cubic surfaces
- Intersection of surfaces
- Rational points