Hypersurfaces achieving the Homma-Kim bound

Abstract

Let $X$ be a hypersurface in ${\mathbb{P}}^N$ with $N\ge 3$ 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.

arXiv comment14 pages; the final presentation of the main topics has been improved by some changes and additional results, and some mistakes and typos have been corrected