Patrick Brosnan, Zinovy Reichstein and Angelo Vistoli: Essential dimension and algebraic stacks I., and

Submission: 2007, Oct 31, abstract updated 2009, July 5.

Essential dimension is a numerical invariant of an algebraic group G which reflects the complexity of G-torsors over fields. In the past 10 years it has been studied by many authors in a variety of contexts. In this paper we extend this notion to algebraic stacks. As an application of the resulting theory we obtain new results about the essential dimension of certain algebraic (and, in particular, finite) groups which occur as central extensions. In particular, we show that the essential dimension of the spinor group Spin_n grows exponentially with n. In the last section we apply the result on spinor groups to show that quadratic forms with trivial discriminant and Hasse-Witt invariant are more complex, in high dimensions, than previously expected. This preprint represents a reworking of the parts of our previous posting (no. 238) concerning algebraic groups and quadratic forms. These results have since been further strengthened and streamlined in the paper "Essential dimension of finite p-groups" by Karpenko and Merkurjev (no. 263) and in our papers "Essential dimension, spinor groups and quadratic forms" and "Essential dimension of moduli spaces of curves and other algebraic stacks". See also a related paper "Some consequences of the Karpenko-Merkurjev theorem" by A. Meyer and Z. Reichstein (no. 309).

2000 Mathematics Subject Classification: 14A20, 20G15, 11E04

Keywords and Phrases: Essential dimension, algebraic stack, linear algebraic group, non-abelian cohomology, Pfister form

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