Higher order Petri Loci

TL;DR

The paper addresses higher order Petri loci $\mathcal{P}_{g,d}^{r,k}$, studying when a curve carries a $g^r_d$ with a Petri kernel of dimension $k$ and determining their codimension and smoothness in $\mathcal{M}_g$. It develops a deformation-theoretic framework using the Petri map $\mu_0$ and the Wahl map $\mu_1$, linking tangent spaces to deformations of the bundle and the curve to kernel dimensions. It then constructs smoothable limit linear series on chains of elliptic curves via admissible rectangle fillings, producing explicit kernel elements and proving that, for suitable parameters, one can realize codimension-$k$ loci with prescribed Petri-kernel dimensions, and extends these to components in $\mathcal{M}_g$ of codimension $k$. The method blends degenerations, Brill–Noether theory, and explicit glueing arguments to yield constructive existence results for smooth Petri loci of prescribed codimension, including cases with negative Brill–Noether number. Overall, the work broadens understanding of the geometry of Petri loci and provides a framework to construct and analyze higher-order degeneracy phenomena in the moduli of curves.

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 ${\mathcal P}_{g,d}^{r,k}$.

Figures (1)

• Figure 1:

Theorems & Definitions (16)

• Lemma 1.1
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof