A note on the Petri loci

TL;DR

This paper investigates the structure of Petri loci within the moduli space of complex curves, establishing codimension results under certain Brill-Noether conditions.

Contribution

It proves that components of the Petri locus with general curves lacking certain linear systems have codimension one in the moduli space when the Brill-Noether number is non-negative.

Findings

Petri loci components have codimension one under specified conditions.

The result applies when the Brill-Noether number is non-negative.

General curves in these components lack specific linear systems.

Abstract

Let $\M_g$ be the course moduli space of complex projective nonsingular curves of genus $g$. We prove that when the Brill-Noether number $\rho(g,r,n)$ is non-negative every component of the Petri locus $P^r_{g,n}\subset \M_g$ whose general member is a curve $C$ such that $W^{r+1}_n(C) = \emptyset$, has codimension one in $\M_g$.