Hypersurfaces achieving the Homma-Kim bound

TL;DR

This paper classifies hypersurfaces over finite fields that reach the Homma-Kim bound on rational points, providing a complete understanding of their structure when they lack linear components.

Contribution

It offers a classification of hypersurfaces attaining the Homma-Kim bound without linear components, up to projective equivalence.

Findings

Hypersurfaces achieving the Homma-Kim bound are classified.

The classification is complete up to projective equivalence.

Hypersurfaces with no linear component are characterized precisely.

Abstract

Let $X$ be a hypersurface in $\mathbb{P}^N$ with $N\geq 3$ defined over a finite field. The main result of this note is the classification, up to projective equivalence, of hypersurfaces $X$ as above without a linear component when the number of their rational points achieves the Homma-Kim bound.