The image in Mg of strata of meromorphic and quadratic differentials

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