Combinatorial structure of low degree rational curves on a smooth Hermitian surface

TL;DR

This paper investigates the combinatorial structure of low-degree rational curves on smooth Hermitian surfaces, revealing their connection to strongly regular graphs and association schemes in algebraic geometry.

Contribution

It introduces a novel link between rational curves on Hermitian surfaces and combinatorial structures like strongly regular graphs and association schemes.

Findings

Rational curves of degree q+1 on Hermitian surfaces form strongly regular graphs.

These curves induce association schemes with specific combinatorial properties.

The study enhances understanding of the interplay between algebraic geometry and combinatorics.

Abstract

A smooth Hermitian surface $X$ is a projective surface isomorphic to the Fermat surface of degree $q+1$ in positive characteristic. We study incidence relations of the rational curves of degree $q+1$ contained in $X$, and show that such curves produce a family of certain strongly regular graphs and association schemes.