Singular curves over a finite field and with many points

TL;DR

This paper extends the study of singular projective curves over finite fields with many points, building on recent work that identified extremal properties, and broadens the scope to include more general non-rational curves.

Contribution

It generalizes the definition of extremal singular curves over finite fields to encompass a wider class of non-rational curves, expanding the theoretical framework.

Findings

Extended the class of singular curves with many points over finite fields.

Provided new examples of extremal curves beyond rational cases.

Enhanced understanding of the properties of non-rational singular curves.

Abstract

Recently Fukasawa, Homma and Kim introduced and studied certain projective singular curves over $\mathbb {F}_q$ with many extremal properties. Here we extend their definition to more general non-rational curves.