A note on the Petri loci

Let $\M_g$ be the course moduli space of complex projective nonsingular curves of genus $g$. We prove that when the Brill-Noether number $\rho(g,r,n)$ is non-negative every component of the Petri locus $P^r_{g,n}\subset \M_g$ whose general member is a curve $C$ such that $W^{r+1}_n(C) = \emptyset$, has codimension one in $\M_g$.