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We compute the dimension of the image of the forgetful map from $Z$ to themoduli space of curves, where $Z$ is a connected component of the stratum of$k$-differentials with an assigned partition $\mu$, for the cases when $k=1$with meromorphic partition and $k=2$ when the quadratic differentials have atworst simple poles.

The image in the moduli space of curves of strata of meromorphic and quadratic differentials