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

Abstract

It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in ${\mathbb{P}}^4({\mathbb{F}}_{q^2})$ has at most $d(q^5+q^2)+q^3+1$ points in common with a threefold of degree $d$ defined over ${\mathbb{F}}_{q^2}$. He proved the conjecture for $d=2$. In this paper, we show that the conjecture is true for $d=3$ and $q\ge 7$.