The symmetric square of a curve and the Petri map

TL;DR

This paper investigates the structure of the Petri locus in the moduli space of curves, showing it has a divisorial component intersecting the boundary, by relating pencils on curves to curves on symmetric squares.

Contribution

It establishes a connection between the Petri locus and the Hilbert scheme of curves on symmetric squares, providing new insights into the geometry of the moduli space of curves.

Findings

The Petri locus has a divisorial component intersecting the boundary of the moduli space.

The scheme parametrizing degree n pencils is isomorphic to a component of a Hilbert scheme.

Properties of families of curves on symmetric squares are analyzed.

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 $\rho(g,1,n)$ is non-negative the Petri locus $P^1_{g,n}\subset \M_g$ has a divisorial component whose closure has a non-empty intersection with $\Delta_0$. In order to prove the result we show that the scheme $G^1_n(\Gamma)$ that parametrizes degree $n$ pencils on a curve $\Gamma$ is isomorphic to a component of the Hilbert scheme parametrizing certain curves on the symmetric square $\Gamma_2$ of $\Gamma$ and we study the properties of such a family of curves.