Higher order Petri Loci

Denote by ${\mathcal P}_{g,d}^{r,k}$ the subset of the moduli space of curves of genus g consisting of those curves that have a linear series of degree d and dimension r for which the Petri map has kernel of dimension at least k. We show the existence of codimension k components of ${\mathcal P}_{g,d}^{r,k}$.