[Paper Review] A tour of stable reduction with applications

Sebastian Casalaina‐Martin|arXiv (Cornell University)|Jul 4, 2012

Algebraic Geometry and Number Theory24 references3 citations

TL;DR

This survey presents a comprehensive overview of stable reduction theorems across algebraic geometry, focusing on curves, abelian varieties, and canonically polarized varieties. It establishes that after a finite base change, families over the punctured disk can be uniquely extended to stable families over the disk, providing a modular compactification of moduli spaces and unifying results via the valuative criterion of properness for Deligne–Mumford stacks.

ABSTRACT

The Deligne-Mumford stable reduction theorem asserts that for a family of stable curves over the punctured disk, after a finite base change, the family can be completed in a unique way to a family of stable curves over the disk. In this survey we discuss stable reduction theorems in a number of different contexts. This includes a review of recent results on abelian varieties, canonically polarized varieties, and singularities. We also consider the semi-stable reduction theorem and results concerning simultaneous stable reduction.

Motivation & Objective

• To extend the classical stable reduction theorem for curves to broader geometric contexts, including abelian varieties and canonically polarized varieties.
• To clarify the role of semi-stable reduction as a foundational tool in constructing stable reductions, particularly in characteristic zero.
• To address simultaneous stable reduction problems for higher-dimensional bases, motivated by birational geometry and moduli theory.
• To compare different stability conditions—especially GIT and Deligne–Mumford stability—via explicit examples like plane quartics.
• To connect stable reduction to the Hassett–Keel program, showing how different compactifications arise from distinct linearizations and stability conditions.

Proposed method

• Utilizes the valuative criterion of properness for moduli stacks as a unifying framework for stable reduction.
• Applies semi-stable reduction via Mumford's theorem, ensuring existence of smooth total spaces with simple normal crossing central fibers after base change.
• Employs log canonical models of semi-stable reductions to construct stable compactifications, particularly in the context of Kollár–Shepherd-Barron–Alexeev theory.
• Uses GIT constructions, especially Gieseker's Hilbert scheme quotient approach, to realize moduli spaces of stable curves as quotients.
• Analyzes degenerations of plane quartics to compare GIT-stable and Deligne–Mumford-stable central fibers, highlighting differences in central fiber structure.
• Applies stack-theoretic methods and rational maps between moduli spaces to resolve birational maps, particularly in the Hassett–Keel program.

Experimental results

Research questions

• RQ1 Under what conditions can a family of stable curves over the punctured disk be uniquely extended to a family over the disk after a finite base change?
• RQ2 How do different stability conditions (e.g., GIT vs. Deligne–Mumford) lead to distinct central fibers in stable reduction, and what are the geometric implications?
• RQ3 What is the role of semi-stable reduction in constructing stable reductions, and why is it not unique?
• RQ4 Can simultaneous stable reduction be achieved for families over higher-dimensional bases, and what are the obstructions?
• RQ5 How do the moduli spaces $ί{M}_3^{GIT}$ and $ί{M}_3$ relate birationally, and what does this imply for the Hassett–Keel program?

Key findings

• The stable reduction theorem ensures that after a finite base change, a family of stable curves over the punctured disk extends uniquely to a family over the disk with a stable central fiber.
• Semi-stable reduction exists in characteristic zero for all families, but the completion is not unique, and the central fiber has simple normal crossing singularities.
• For plane quartics degenerating to a tacnodal curve, two distinct stable reductions arise: one as a union of two conics meeting at two points, and another as an elliptic bridge (two elliptic curves meeting at two points).
• The moduli space $ί{M}_g$ of Deligne–Mumford stable curves is isomorphic to a GIT quotient of a Hilbert scheme via Gieseker's construction, providing an alternative proof of stable reduction.
• The space $ί{M}_3^{GIT}$ is birational to $ί{M}_3$, with the rational map induced by families of smooth curves with trivial automorphism group.
• Resolving the rational map $ί{M}_3^{GIT} o ί{M}_3$ is deeply connected to simultaneous stable reduction for curves with $ADE$ singularities, as studied in the Hassett–Keel program.