The characterization of Hermitian surfaces by the number of points

TL;DR

This paper characterizes nonsingular Hermitian surfaces over finite fields by their number of rational points, distinguishing them among surfaces of the same degree in projective 3-space.

Contribution

It provides a unique characterization of Hermitian surfaces based on point counts, enhancing understanding of their geometric and arithmetic properties.

Findings

Hermitian surfaces are uniquely identified by their number of ${F}_q$-points.

The characterization applies to surfaces of degree $ oot{q} +1$ in projective 3-space.

This result aids in classifying algebraic surfaces over finite fields.

Abstract

The nonsingular Hermitian surface of degree $\sqrt{q} +1$ is characterized by its number of $\Bbb{F}_q$-points among the irreducible surfaces over $\Bbb{F}_q$ of degree $\sqrt{q} +1$ in the projective 3-space.