Double Points of Plane Models in M_{g,1}

TL;DR

This paper computes the class of a special divisor in the moduli space of genus 6 pointed curves, revealing its extremal nature in the pseudoeffective cone, and introduces a general method for certain linear series.

Contribution

It provides the explicit class of a divisor in M_{6,1} and introduces a new approach for families of linear series with specific Brill-Noether numbers.

Findings

The divisor generates an extremal ray in the pseudoeffective cone.

A general result on linear series with Brill-Noether number 0 or -1 is established.

The class of the divisor is explicitly computed.

Abstract

The aim of this paper is to compute the class of the closure of the effective divisor in M_{6,1} given by pointed curves [C,p] with a sextic plane model mapping p to a double point. Such a divisor generates an extremal ray in the pseudoeffective cone of M_{6,1} as shown by Jensen. A general result on some families of linear series with adjusted Brill-Noether number 0 or -1 is introduced to complete the computation.