The symmetric square of a curve and the Petri map

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 $G_n^1(\Gamma )$ 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.