Essential dimension of moduli of curves and other algebraic stacks
Patrick Brosnan, Zinovy Reichstein, Angelo Vistoli, Najmuddin Fakhruddin
备注：
36 pages. The main change is the addition of an appendix by Najmuddin Fakhruddin, containing a complete answer for the question of the essential dimension of the stack of principally polarized abelian varieties
发表信息：
J. European Math. Society (JEMS) 13 (2011), no. 4, 1079-1112
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).
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