The Abelian Monodromy Extension Property for Families of Curves

TL;DR

This paper introduces the abelian monodromy extension (AME) property, providing necessary and sufficient conditions for extending families of smooth curves to stable curves, and characterizes unique compactifications of moduli spaces.

Contribution

It defines the AME property and proves the Deligne-Mumford and Baily-Borel compactifications are uniquely maximal AME compactifications.

Findings

Deligne-Mumford compactification is the unique maximal AME compactification of the moduli space of curves.

Baily-Borel compactification is the unique maximal projective AME compactification of the moduli space of abelian varieties.

Provides monodromy-based criteria for extending families of curves over open subsets.

Abstract

Necessary and sufficient conditions are given (in terms of monodromy) for extending a family of smooth curves over an open subset U of S to a family of stable curves over S. More precisely, we introduce the abelian monodromy extension (AME) property and show that the standard Deligne-Mumford compactification is the unique, maximal AME compactification of the moduli space of curves. We also show that the Baily-Borel compactification is the unique, maximal projective AME compactification of the moduli space of abelian varieties.