Donghoon Hyeon

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

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.

STABILITY COMPUTATION VIA GROBNER BASIS

In this article, we discuss a Grobner basis algorithm related to the stability of algebraic varieties in the sense of Geometric Invariant Theory. We implement the algorithm with Macaulay 2 and use it to prove the stability of certain curves that play an important role in the log minimal model program for the moduli space of curves.