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

TL;DR

This paper proves Edoukou's conjecture that a non-degenerate Hermitian threefold in projective 4-space over a finite field has a maximum number of intersection points with a degree 3 threefold, for certain parameters.

Contribution

The paper confirms Edoukou's conjecture for degree 3 threefolds over finite fields with q ≥ 7, extending previous results beyond the quadratic case.

Findings

Confirmed the conjecture for d=3 and q ≥ 7

Established an upper bound on intersection points

Extended known results to higher degree threefolds

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 $q \ge 7$.