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In this paper we address questions of the following type. Let k be a basefield 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 afield of definition of $X$ over the base field k? In other words, how manyindependent parameters are needed to define X? To study these questions weintroduce a notion of essential dimension for an algebraic stack. Inparticular, we give a complete answer to the question above when the geometricobjects X are smooth or stable curves, and, in the appendix by N. Fakhruddin,for principally polarized abelian varieties.  This paper overlaps with our earlier preprint arXiv:math/0701903 . Thatpreprint has splintered into several parts, which have since acquired a life oftheir own. In particular, see "Essential dimension, spinor groups, andquadratic forms", by the same authors, and "Some consequences of theKarpenko-Merkurjev theorem", by Meyer and Reichstein (arXiv:0811.2517).

Essential dimension of moduli of curves and other algebraic stacks