Uniruledness of Some Moduli Spaces of Pointed Spin Curves

The moduli space ${\mathcal{S}}_{g,2n}$ parametrizes pointed curves with spin structure. We prove that ${\mathcal{S}}_{g,2}$, ${\mathcal{S}}_{g,4}$ and ${\mathcal{S}}_{g,6}$ are uniruled for particular values of $g$.