Towards a classification of modular compactifications of M_{g,n}

Abstract

The moduli space of smooth curves admits a beautiful compactifi- cation Mg,n ? Mg,n by the moduli space of stable curves. In this paper, we undertake a systematic classification of alternate modular compactifications of Mg,n. Let Ug,n be the (non-separated) moduli stack of all n-pointed reduced, connected, complete, one-dimensional schemes of arithmetic genus g. When g = 0, U0,n is irreducible and we classify all open proper substacks of U0,n. When g = 1, Ug,n may not be irreducible, but there is a unique irreducible component Vg,n ? Ug,n containing Mg,n. We classify open proper substacks of Vg,n satisfying a certain stability condition.