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 $ρ(g,1,n)$ 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 $Δ_0$. In order to prove the result we show that the scheme $G^1_n(Γ)$ that parametrizes degree $n$ pencils on a curve $Γ$ is isomorphic to a component of the Hilbert scheme parametrizing certain curves on the symmetric square $Γ_2$ of $Γ$ and we study the properties of such a family of curves.

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