Log minimal model program for the moduli space of stable curves

Abstract

We give a geometric invariant theory (GIT) construction of the log canonical model Mg( of the pairs (Mg; for (7=10 ; 7=10] for small 2 Q+. We show that Mg(7=10) is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; Mg(7=10 ) is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction