Log canonical models for the moduli space of curves: The first divisorial contraction

Abstract

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.

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

“…In the case of genus 3 curves, in the general context of searching for the canonical model of M g (see Hassett-Hyeon [40]), one has a quite precise understanding of the birational geometry of M 3 (see Hyeon-Lee [41] and Rulla [56]). It is shown in Hyeon-Lee [41] that the birational models of M 3 mentioned above are log canonical models for an appropriate choice of the boundary.…”

Stability Computation via Grobner Basis

“…Moreover, considering the functoriality of the Hilbert-Mumford index and the tautological ring of the Hilbert scheme reveals that one only needs to compute the Hilbert-Mumford index for mth Hilbert points for finitely many m to obtain the Hilbert-Mumford indices for all m. The results in the remainder of the section are taken from [5].…”

“…These examples played important roles in our work on moduli problems of ν-canonical curves for ν = 2, 3 [7,8,6,5].…”

The global geometry of the moduli space of curves

“…[SB]). Precisely, N E 1 (A I g ) is generated by two curve classes C 1 and C 2 , where C 1 is any exceptional curve in the contraction of A I g to the Satake compactification of A g , while (see [Ha1], [Ha2]). Precisely, if δ := δ 0 + · · · + δ [g/2] denotes the total boundary of M g and K M g = 13λ − 2δ is the canonical class of the moduli stack, for each rational parameter 0 ≤ α ≤ 1, one can introduce the log canonical model…”