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.
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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. 1079–1112DOI 10.4171/JEMS/276