# Essential dimension of moduli of curves and other algebraic stacks

### Zinovy Reichstein

University of British Columbia, Vancouver, Canada### Patrick Brosnan

University of British Columbia, Vancouver, Canada### Angelo Vistoli

Scuola Normale Superiore, Pisa, Italy

## Abstract

In this paper we consider questions of the following type. Let *k* be a base field and *K*/*k* be a field extension. Given a geometric object *X* over a field *K* (e.g. a smooth curve of genus *g*) what is the least transcendence degree of a field of definition of *X* over the base field *k*? In other words, how many independent parameters are needed to define *X*? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete answer to the question above when the geometric objects *X* are smooth, stable or hyperelliptic curves. The appendix, written by Najmuddin Fakhruddin, answers this question in the case of abelian varieties.

## Cite this article

Zinovy Reichstein, Patrick Brosnan, Angelo Vistoli, Essential dimension of moduli of curves and other algebraic stacks. J. Eur. Math. Soc. 13 (2011), no. 4, pp. 1079–1112

DOI 10.4171/JEMS/276