Maximum number of ${\mathbb{F}}_q$-rational points on nonsingular threefolds in ${\mathbb{P}}^4$

Abstract

We determine the maximum number of ${\mathbb{F}}_q$-rational points that a nonsingular threefold of degree $d$ in a projective space of dimension $4$ defined over ${\mathbb{F}}_q$ may contain. This settles a conjecture by Homma and Kim concerning the maximum number of points on a hypersurface in a projective space of even dimension in this particular case.

8 pages