A note on the Petri loci (Q644992)

A smooth projective curve \(C\) is called a Petri curve if for any line bundle \(L\) on \(C\) the Petri map \[ H^0(L)\otimes H^0(\omega_C\otimes L^{-1})\longrightarrow H^0(\omega_C) \] is injective. By the Gieseker-Petri theorem, the Petri locus \(P_g\) consisting of curves of genus \(g\) which are not Petri is a proper closed subset of \(M_g\). Denote \(P^r_{g,n}\subset M_g\) the locus of curves which are not Petri w.r.t \(g^r_n\). It is conjectured that \(P^r_{g,n}\) has pure codimension one in \(M_g\) if the Brill-Noether number \(\rho(g,r,n)\geq0\). The authors proved a special case of the conjecture, namely that if \(\rho(g,r,n)\geq0\) then every component of \(P^r_{g,n}\) whose general member is a curve \(C\) with \(W^{r+1}_n(C)=\emptyset\) has codimension one in \(M_g\).