Uniruledness of some moduli spaces of stable pointed curves

We prove uniruledness of some moduli spaces of stable curves of genus g with n marked points using linear systems on nonsingular projective surfaces containing the general curve of genus g. Precisely we show that is uniruled for g=12 and n≤5, g=13 and n≤3, g=15 and n≤2.We then prove that the pointed hyperelliptic locus Hg,n is uniruled for g≥2 and n≤4g+4.In the last part we show that a nonsingular complete intersection surface does not carry a linear system containing the general curve of genus g≥16 and if it carries a linear system containing the general curve of genus 12≤g≤15, then it is canonical.