Rational curves on a smooth Hermitian surface II: moduli and counting of certain infinite families

A smooth $k$-Hermitian surface $X$ is a surface projectively isomorphic over $k$ to the Fermat surface of degree $q+1$, where $k$ is an algebraic closure of a finite field with $q^2$ elements.

We consider the curves on $X$ parametrized by polynomials with exactly $4$ terms under a suitable choice of the parameter.

We determine the $k$-projective equivalence classes and the moduli spaces of such curves of all degrees.

It is shown that each equivalence class has only one orbit under the action of the group of the automorphisms of $X$ if $q\ge3$ while if $q=2$ all the classes split into infinitely many orbits, and moreover, the moduli spaces are affine algebraic sets.

We also determine the number of the curves belonging to each orbit.

In addition, the smoothness, the reflexivity and the minimal field of definition for such curves are shown.