[Paper Review] Weakly proper moduli stacks of curves

Jarod Alper, David Ishii Smyth+1 more|arXiv (Cornell University)|Dec 2, 2010

Algebraic Geometry and Number Theory4 references18 citations

TL;DR

This paper introduces the concept of weakly proper algebraic stacks to construct modular compactifications of the moduli space of curves without relying on Geometric Invariant Theory (GIT). It defines and proves the weak properness of moduli stacks parameterizing curves with $A_k^-$, $A_k$, and $A_k^+$ singularities—specifically nodes, cusps, tacnodes, and ramphoid cusps—laying the foundation for constructing log canonical models of $\overline{M}_g$ via good moduli spaces.

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 $\bar{M}_g$.

Motivation & Objective

• To define and study weakly proper algebraic stacks as a framework for moduli problems with mild non-separatedness, common in GIT.
• To construct moduli stacks $\overline{\mathcal{M}}_{g,n}(A_k^-)$, $\overline{\mathcal{M}}_{g,n}(A_k)$, and $\overline{\mathcal{M}}_{g,n}(A_k^+)$ parameterizing curves with increasingly singular $A_k$-singularities.
• To prove that these stacks are weakly proper, enabling the construction of projective good moduli spaces in subsequent work.
• To provide a GIT-free approach to the Hassett-Keel log minimal model program for $\overline{M}_g$, particularly for the second flip.
• To extend the modular interpretation of log canonical models to the case of $\overline{M}_{g,n}$ with $n > 0$.

Proposed method

• Introduces the notion of weak properness as an abstract characterization of moduli stacks that are mildly non-separated but still admit proper moduli spaces.
• Uses deformation theory and local variation of GIT to analyze the behavior of families of curves with $A_k$-singularities over a punctured disc.
• Applies the theory of good moduli spaces (from Alper [2008]) to show that weakly proper stacks admit projective good moduli spaces.
• Employs base change and cartesian diagrams to lift families of curves to central fibers, proving existence and uniqueness of limits.
• Leverages weak separation and étale morphisms to establish uniqueness of closed $A_k^+$-stable limits under base change.
• Uses the structure of $A_k$-singularities (nodes, cusps, tacnodes, ramphoid cusps) to define $A_k^-$, $A_k$, and $A_k^+$-stability conditions.

Experimental results

Research questions

• RQ1 Can weakly proper algebraic stacks serve as a viable alternative to GIT for constructing moduli spaces of curves with controlled singularities?
• RQ2 Do moduli stacks parameterizing curves with $A_k^-$, $A_k$, and $A_k^+$ singularities satisfy weak properness without relying on semistability analysis?
• RQ3 Can the second flip in the Hassett-Keel log minimal model program for $\overline{M}_g$ be constructed via weakly proper stacks and good moduli spaces?
• RQ4 Is there a modular interpretation of log canonical models $\overline{M}_g(\alpha)$ without using GIT?
• RQ5 Can the construction be extended to $\overline{M}_{g,n}$ with $n > 0$?

Key findings

• The moduli stacks $\overline{\mathcal{M}}_{g,n}(A_k^-)$, $\overline{\mathcal{M}}_{g,n}(A_k)$, and $\overline{\mathcal{M}}_{g,n}(A_k^+)$ are proven to be weakly proper for $k \in \{2,3,4\}$, satisfying the existence and uniqueness of limits under base change.
• The uniqueness of closed $A_k^+$-stable limits is established via specialization arguments and weak separation in the stack $\overline{\mathcal{M}}_{g,n}(A_k)$.
• The construction of good moduli spaces for these stacks is shown to be possible in forthcoming work, relying on weak properness and the theory of good moduli spaces.
• The stacks $\overline{\mathcal{M}}_{g,n}(A_k)$ are shown to be weakly separated, a key ingredient in proving uniqueness of limits.
• The framework enables a GIT-free construction of log canonical models for $\overline{M}_g$, with the second flip in the Hassett-Keel program as the ultimate goal.
• The results lay the foundation for extending modular compactifications to $\overline{M}_{g,n}$ with $n > 0$.