Abstract. In this paper, we initiate our investigation of log canonical models for (M g , αδ) as we decrease α from 1 to 0. We prove that for the first critical value α = 9/11, the log canonical model is isomorphic to the moduli space of pseudostable curves, which have nodes and cusps as singularities. We also show that α = 7/10 is the next critical value, i.e., the log canonical model stays the same in the interval (7/10, 9/11]. In the appendix, we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space.

Abstract. 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 − ǫ, 7/10] for small ǫ ∈ Q + . We show that M g (7/10) is isomorphic to the GIT quotient of the Chow variety bicanonical curves; M g (7/10 − ǫ) 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 bridges. We show that there is a small contraction Ψ : M g (7/10 + ǫ) → 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 − ǫ) → M g (7/10) that is the Mori flip of Ψ.

In this paper, we completely work out the log minimal model program for the moduli space of stable curves of genus three. We employ a rational multiple αδ of the divisor δ of singular curves as the boundary divisor, construct the log canonical model for the pair (M 3 , αδ) using geometric invariant theory as we vary α from one to zero, and give a modular interpretation of each log canonical model and the birational maps between them. By using the modular description, we are able to identify all but one log canonical models with existing compactifications of M 3 , some new and others classical, while the exception gives a new modular compactification of M 3 . ′ 4 31 4.3. P 4 and P ′ 4 as log canonical models 32 4.4. Kondo's compact moduli space 33 References 34 1 M 3 (1) ≃ M 3 T M 3 ( 9 11 ) ≃ M ps 3 Ψ M 3 (7/10) ≃ M cs 3 M 3 ( 17 28 ) ≃ Q * This program was initiated by Brendan Hassett and Sean Keel, and the ideas were further developed in [HH06]. The genus two case was completely worked out by Hassett in [Has05] (see also [HL07]) and the first couple of steps of the program for the higher genera case were completed in [HH06] and [HH07].We work over an algebraically closed field k of characteristic zero. Acknowledgement. D.H. would like to thank Brendan Hassett for suggesting this problem and for many helpful conversations. He was partially supported by KIAS. He gratefully acknowledges Bumsig Kim for the hospitality and for useful conversations.Y.L. would like to thank Shigeyuki Kondo for his explanation of his compact moduli space, Shigefumi Mori for his explanation of birational geometry, Shigeru Mukai for his valuable comments on the geometric invariant theory.

We show that the GIT quotients of suitable loci in the Hilbert and Chow schemes of 4-canonically embedded curves of genus g ≥ 3 are the moduli space M ps g of pseudostable curves constructed by Schubert in [9] using Chow varieties and 3-canonical models. The only new ingredient needed in the Hilbert scheme variant is a more careful analysis of the stability with respect to a certain 1-ps λ of the m th Hilbert points of curves X with elliptic tails. We compute the exact weight with which λ acts, and not just the leading term in m of this weight. A similar analysis of stability of curves with rational cuspidal tails allows us to determine the stable and semistable 4-canonical Chow loci. Although here the geometry of the quotient is more complicated because there are strictly semistable orbits, we are able to again identify it as M ps g . Our computations yield, as byproducts, examples of both m-Hilbert unstable and m-Hilbert stable X that are Chow strictly semistable.