Let $\mathcal{M}_g$ be the moduli space of genus $g$ curves and $\mathcal{M}_{g,d}^r = \{[C] \in \mathcal{M}_g| \text{ C carries a } g_d^r \}$ the locus of genus $g$ curves carrying a linear system of degree $d$ and dimension $r$. Then for $r = 1$ this is an irreducible subvariety of dimension $2g + 2d - 5$. Is the dimension of $\mathcal{M}_{g,d}^r$ known for arbitrary $r$?
Update: So I found a paper where it is stated, that for negative Brill Noether number $\rho = \rho(g,d,r)$ (for $\rho \geq 0$, $\mathcal{M}_{g,d}^r = \mathcal{M}_{g}$), the codimension of $\mathcal{M}_{g,d}^r$ is expected to be $- \rho$ and this is known to be true for $\rho \in \{-1, -2, -3\}$. It also seems to be known (but I didn't find a reference), that $\mathcal{M}_{g,d}^r$ is not always of pure dimension. Does anyone know of an example, where either all or at least some components of $\mathcal{M}_{g,d}^r$ fail to be of this expected dimension?