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

TL;DR

This paper proves Sorensen's conjecture for the maximum number of intersection points between cubic surfaces and Hermitian surfaces over finite fields, and classifies the surfaces achieving these bounds, challenging previous conjectures.

Contribution

It extends the proof of Sorensen's conjecture to degree 3 surfaces for q ≥ 8 and classifies all such surfaces with maximum and second maximum intersections, disproving prior conjectures.

Findings

Confirmed Sorensen's conjecture for degree 3 surfaces when q ≥ 8.

Identified the second highest number of intersection points.

Classified all cubic surfaces with extremal intersection properties.

Abstract

In 1991 S{\o}rensen 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 $\PP^3(\Fqt)$. 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 $d=3$ and $q \ge 8$. We further 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 second highest number of points in common with a non-degenerate Hermitian surface. This classifications disproves one of the conjectures proposed by Edoukou, Ling and Xing.