The image in $${\mathcal {M}}_g$$ of strata of meromorphic and quadratic differentials

We compute the dimension of the image of the map $$\pi _Z :Z \rightarrow {\mathcal {M}}_g$$ π Z : Z → M g forgetting the markings, where Z is a connected component of the stratum $${\mathcal {H}}^k_g(\mu )$$ H g k ( μ ) of k -differentials with an assigned partition $$\mu $$ μ , for the cases when $$k=1$$ k = 1 with meromorphic partition and $$k=2$$ k = 2 when the quadratic differentials have at worst simple poles.