Extremal higher codimension cycles on moduli spaces of curves.

We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space Mg,n of stable genus g curves with n ordered marked points. In particular, we prove that codimension two boundary strata are extremal and exhibit extremal boundary strata of higher codimension. We also show that the locus of hyperelliptic curves with a marked Weierstrass point in M3,1 and the locus of hyperelliptic curves in M4 are extremal cycles. In addition, we exhibit infinitely many extremal codimension two cycles in M1,n for n ≥ 5 and in M2,n for n ≥ 2.