Intersection-theoretical computations on \Mgbar

We determine necessary conditions for ample divisors in arbitrary genus as well as for very ample divisors in genus 2 and 3. We also compute the intersection numbers $\lambda^9$ and $\lambda_{g-1}^3$ in genus 4. The latter number is relevant for counting curves of higher genus on manifolds, cf. the recent work of Bershadsky et al.