Nondegenerate curves of low genus over small finite fields

TL;DR

This paper classifies nondegenerate curves of genus up to 3 over small finite fields, identifying exactly two exceptions with notable extremal properties regarding rational points.

Contribution

It proves that only two genus ≤3 curves over small finite fields are non-nondegenerate, highlighting their unique extremal characteristics.

Findings

Exactly two genus ≤3 curves over small fields are non-nondegenerate.

Both exceptional curves have extremal rational point counts.

The result contrasts with large field cases where all such curves are nondegenerate.

Abstract

In a previous paper, we proved that over a finite field $k$ of sufficiently large cardinality, all curves of genus at most 3 over k can be modeled by a bivariate Laurent polynomial that is nondegenerate with respect to its Newton polytope. In this paper, we prove that there are exactly two curves of genus at most 3 over a finite field that are not nondegenerate, one over F_2 and one over F_3. Both of these curves have remarkable extremal properties concerning the number of rational points over various extension fields.