  In this paper we address 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. In
particular, we give a complete answer to the question above when the geometric
objects 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 . That
preprint has splintered into several parts, which have since acquired a life of
their own. In particular, see "Essential dimension, spinor groups, and
quadratic forms", by the same authors, and "Some consequences of the
Karpenko-Merkurjev theorem", by Meyer and Reichstein (arXiv:0811.2517).
