Counting the number of trigonal curves of genus 5 over finite fields

TL;DR

This paper develops a counting method for trigonal genus 5 curves over finite fields using plane quintics with a singularity, enabling computation of the moduli space's motivic Euler characteristic.

Contribution

It introduces a novel counting approach for trigonal genus 5 curves via plane quintics with a specific singularity, linking geometric properties to arithmetic counts.

Findings

Count of trigonal genus 5 curves over finite fields obtained

Motivic Euler characteristic of the moduli space computed

Method applicable to other curve counting problems

Abstract

The trigonal curves of genus 5 can be represented by projective plane quintics that have 1 singularity of delta invariant 1. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this then gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.