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

2018/02/19 by Beelen, Peter, Datta, Mrinmoy
#05B25 #14G05 #14G15 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1802.06681

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 \PP³(\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 ≥ 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.

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