A monodromy criterion for extending curves

TL;DR

This paper establishes a criterion based on monodromy for determining when a family of smooth curves of genus at least 2 extends over a base variety, using fundamental group isomorphisms.

Contribution

It provides a more precise monodromy criterion for extending families of curves over a base variety, refining previous understanding.

Findings

Extension of curves characterized by fundamental group isomorphism

Provides a necessary and sufficient monodromy criterion

Applicable to families parametrized by dense open subsets

Abstract

A family of proper smooth curves of genus $\geq 2$, parametrised by an open dense subset $U$ of a normal variety $S$, extends to $S$ if the natural map $\pi_1(U) \to \pi_1(S)$ on fundamental groups is an isomorphism. The criterion of this note is actually more precise.