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

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.

💡 Research Summary

The paper investigates the combinatorial structures arising from rational curves of degree q + 1 that lie on a smooth Hermitian surface X over an algebraically closed field k of characteristic p, where X is projectively equivalent to the Fermat surface defined by x₀^{q+1}+x₁^{q+1}+x₂^{q+1}+x₃^{q+1}=0. The automorphism group of X, Aut(X) ≅ PGU₄(F_{q²}), acts transitively on three natural sets: the F_{q²}-rational points X(F_{q²}), the set of lines L(X), and, for q ≥ 3, the set R(X) of all rational curves of degree q + 1 contained in X. The cardinalities are |X(F_{q²})| = (q³ + 1)(q² + 1), |L(X)| = (q³ + 1)(q + 1), and |R(X)| = q⁴(q³ + 1)(q² − 1).

Choosing a concrete representative curve C_F (defined by an explicit parametrisation involving a matrix F_J) the authors show that every curve in R(X) is an image of C_F under Aut(X). They then fix an Aut(X)-orbit O = Aut(X)·C_F and study its incidence with the point set V = X(F_{q²}). Defining E = {(v₁,v₂) ∈ V×V | v₁ and v₂ lie on a common curve C ∈ O}, they prove that (V,E) is a rank‑3 strongly regular graph with parameters

• v = (q³ + 1)(q² + 1) (the number of vertices),
• k = q⁵ (regular degree),
• λ = q(q − 1)(q³ + q² − 1),
• μ = q³(q² − 1).

These graphs turn out to be the complements of the point graphs of the generalized quadrangle GQ(q²,q). For q = 2 and q = 3 the authors compute the explicit parameters: (45, 32, 22, 24) and (280, 243, 210, 216) respectively, and identify the full automorphism groups as 51840·PSU₄(2) and 26127360·PSU₄(3).

Next, the paper analyses intersections of distinct curves in O. Theorem 4.3 establishes that any two distinct curves intersect in at least one point; indeed, for q ≥ 3 all curves are defined over F_{q²} so each contains an F_{q²}-rational point, and the stabiliser arguments guarantee a non‑empty intersection. Consequently, the set of possible intersection numbers {m₁,…,m_d} is finite and non‑zero. Using these numbers they define relations R_i on O by “two curves intersect in exactly m_i points”. The pair (O,{R_i}) becomes a d‑class symmetric association scheme. For q = 2 they obtain d = 5 with intersection numbers {1,2,3,4,5} and present the eigenmatrices P and Q, as well as the intersection matrices L_i and their duals L_i^*. For q = 3 they find d = 10 with intersection numbers {1,2,3,4,5,6,7,8,10,20} and give analogous data. The authors observe that d equals |P¹(F_{q²})| = q² + 1 in the computed cases and conjecture this holds for all q.

In Section 5 the authors construct Schurian schemes from the diagonal action of Aut(X) on the sets X(F_{q²}), L(X), and O. The point set and line set each give a 2‑class scheme corresponding respectively to the point graph and line graph of GQ(q²,q). The curve set O for q = 2 yields a 19‑class non‑commutative scheme; the paper supplies the full character table, eigenvalues, valencies, and ranks of primitive idempotents, illustrating the richer algebraic structure beyond the commutative case.

Overall, the work demonstrates that low‑degree rational curves on a smooth Hermitian surface generate highly symmetric combinatorial objects: families of strongly regular graphs intimately linked to generalized quadrangles, and a hierarchy of association schemes whose parameters are explicitly computable. The transitivity of the large unitary automorphism group underpins all constructions, and the paper opens several avenues for further research, including a proof of the conjectured class number d = q² + 1, deeper analysis of the intersection patterns, and potential applications to coding theory and finite geometry.