Uniruledness of strata of holomorphic differentials in small genus

We address the question concerning the birational geometry of the strata of holomorphic and quadratic differentials. We show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on K3 and del Pezzo surfaces respectively. Restricting to genus 3≤g≤6, we construct projective bundles over a rational varieties that dominate the holomorphic strata with length at most g−1, hence showing in addition that these strata are unirational.