Composantes connexes et irr\'eductibles en familles

TL;DR

This paper introduces and studies notions of relative connected and irreducible components in algebraic stacks, focusing on their representability and applications to moduli stacks of curves with group actions.

Contribution

It defines various notions of relative components in algebraic stacks and analyzes their representability, applying results to moduli stacks of curves with finite group actions.

Findings

Different notions of relative components are introduced and compared.

Conditions for the representability of associated functors are established.

Application to moduli stacks of curves with group actions demonstrates practical relevance.

Abstract

For an algebraic stack $\sX$ flat and of finite presentation over a scheme $S$, we introduce various notions of {\em relative connected components} and {\em relative irreducible components}. The main distinction between these notions is whether we require the total space of a relative component to be open or closed in $\sX$. We study the representability of the associated functors of relative components, and give an application to the moduli stack of curves of genus $g$ admitting an action of a fixed finite group $G$.