Log minimal model program for the moduli space of stable curves: the first flip
Hassett, Brendan; Hyeon, Donghoon
We give a geometric invariant theory (GIT) construction of the log canonical model M¯g(α) of the pairs (M¯g,αδ) for α∈(7/10–ϵ,7/10] for small ϵ∈Q+. We show that M¯g(7/10) is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; M¯g(7/10−ϵ) 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 Ψ:M¯g(7/10+ϵ)→M¯g(7/10) that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction Ψ+:M¯g(7/10−ϵ)→M¯g(7/10) that is the Mori flip of Ψ.