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Submission: 2009, Jul 5
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. This paper represents a reworking of the geometric parts of our earlier preprint "Essential dimension and algebraic stacks" (no. 238). New material has been added, and many of the proofs have been streamlined.
2000 Mathematics Subject Classification: 14A20, 14H10, 11E04
Keywords and Phrases: Essential dimension, algebraic stack, gerbe, genericity theorem, moduli space, abelian variety
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