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

## TL;DR

This paper determines the maximum number of rational points on nonsingular threefolds in projective 4-space over finite fields, settling a specific conjecture for hypersurfaces of even dimension.

## Contribution

It provides a definitive answer to the maximum number of rational points on nonsingular threefolds, confirming a conjecture by Homma and Kim.

## Key findings

- Maximum number of points for given degree d
- Resolution of Homma and Kim's conjecture
- Specific bounds for threefolds in 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.