# Moscow Mathematical Journal

Volume 20, Issue 3, July–September 2020  pp. 453–474.

Maximum Number of Points on Intersection of a Cubic Surface and a Non-Degenerate Hermitian Surface

Authors:  Peter Beelen (1) and Mrinmoy Datta (1)
Author institution:(1) Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK 2800, Kgs. Lyngby, Denmark

Summary:

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 $\mathbb{P}^3(\mathbb{F}_{q^2})$. 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 $q \ge 4$, 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 $q \ge 4$, 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.

2010 Math. Subj. Class. Primary: 14G05, 14G15, 05B25

Keywords: Hermitian surfaces, cubic surfaces, intersection of surfaces, rational points.