Higher order Petri Loci

TL;DR

This paper investigates the structure of certain loci within the moduli space of curves, demonstrating the existence of components with specific codimension related to Petri map kernels.

Contribution

It establishes the existence of codimension k components of the loci where the Petri map's kernel dimension is at least k in the moduli space of curves.

Findings

Existence of codimension k components in ${ m P}_{g,d}^{r,k}$

Identification of loci with prescribed Petri map kernel dimensions

Advancement in understanding the geometry of moduli spaces

Abstract

Denote by ${\mathcal P}_{g,d}^{r,k}$ the subset of the moduli space of curves of genus g consisting of those curves that have a linear series of degree d and dimension r for which the Petri map has kernel of dimension at least k. We show the existence of codimension k components of ${\mathcal P}_{g,d}^{r,k}$.