  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.
