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

TL;DR

This work studies incidence relations among rational curves of degree $q+1$ on a smooth Hermitian surface $X$, identifying a rich family of combinatorial structures arising from the ${ m Aut}(X)$-orbits of these curves. By focusing on the orbit $O={ m Aut}(X_I)C_J$ and the incidence with $X_I( olinebreak[4]{}_{q^2})$, the authors construct strongly regular graphs that are complements of the point graphs of ${ m GQ}(q^2,q)$, and they derive explicit parameters for general $q$. They then define intersection-based relations among curves to obtain $d$-class association schemes on $O$, computing eigenmatrices in low-dimensional cases (e.g., $q=2,3$) and proposing a conjecture that $d=| olinebreak[4]{}_{q^2}{ m P}^1|=q^2+1$ in general. A Schurian perspective for schemes on rational points, lines, and curves is developed, connecting the geometric setup to classical symmetric designs and generalized quadrangles and providing a framework for further exploration of these algebraic-combinatorial structures.

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.

Theorems & Definitions (21)

• Proposition 1.1: BC,C,Se
• Proposition 1.2: O,O2
• Example 3.1
• Example 3.2
• Theorem 3.3
• proof
• Corollary 3.4
• Lemma 3.5
• proof
• Theorem 3.6