The symmetric square of a curve and the Petri map
  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. 
A. Bruno, E. Sernesi
math.AG