Weakly proper moduli stacks of curves

Abstract

This is the first in a projected series of three papers in which we construct the second flip in the log minimal model program for ${\bar{M}}_g$. We introduce the notion of a weakly proper algebraic stack, which may be considered as an abstract characterization of those mildly non-separated moduli problems encountered in the context of Geometric Invariant Theory (GIT), and develop techniques for proving that a stack is weakly proper without the usual semistability analysis of GIT. We define a sequence of moduli stacks of curves involving nodes, cusps, tacnodes, and ramphoid cusps, and use the aforementioned techniques to show that these stacks are weakly proper. This will be the key ingredient in forthcoming work, in which we will prove that these moduli stacks have projective good moduli spaces which are log canonical models for ${\bar{M}}_g$.

66 pages, 3 figures

References in corpus (2)

• Good moduli spaces for Artin stacks
• Moduli spaces of hyperelliptic curves with A and D singularities

Cited by in corpus (12)

• Singularities with G_m-action and the log minimal model program for ${\bar{M}}_g$
• GIT for polarized curves
• A Harder-Narasimhan theory for Kisin modules
• Compactifications of the universal Jacobian over curves with marked points
• GIT Constructions of Log Canonical Models of M_g
• The final log canonical model of the moduli space of stable curves of genus four
• Generalizing the GAGA Principle
• Moduli spaces of hyperelliptic curves with A and D singularities
• An outline of the log minimal model program for the moduli space of curves
• On some modular contractions of the moduli space of stable pointed curves
• GIT Compactifications of M_{0,n} and Flips
• On GIT quotients of Hilbert and Chow schemes of curves