Towards a classification of modular compactifications of Mg,n

Abstract

The moduli space of smooth curves admits a beautiful compactification $$\mathcal{M}_{g,n} \subset \overline{\mathcal{M}}_{g,n}$$ by the moduli space of stable curves. In this paper, we undertake a systematic classification of alternate modular compactifications of $$\mathcal{M}_{g,n}$$. Let $$\mathcal{U}_{g,n}$$ be the (non-separated) moduli stack of all n-pointed reduced, connected, complete, one-dimensional schemes of arithmetic genus g. When g=0, $$\mathcal{U}_{0,n}$$ is irreducible and we classify all open proper substacks of $$\mathcal{U}_{0,n}$$. When g≥1, $$\mathcal{U}_{g,n}$$ may not be ir..