Submission: 2007, Jan 30, revised 2007, Feb 1
We define and study the essential dimension of an algebraic stack. We compute the essential dimension of the stacks M g,n and M g,n of smooth, or stable, n-pointed curves of genus g. We also prove a general lower bound for the essential dimension of algebraic groups with a non-trivial center. Using this, we find new exponential lower bounds for the essential dimension of spin groups and new formulas for the essential dimension of some finite p-groups. Finally, we apply the lower bound for spin groups to the theory of the Witt ring of quadratic forms over a field k.
2000 Mathematics Subject Classification: Primary 14A20; 20G15; 11E04; 14H10
Keywords and Phrases: Essential dimension, algebraic stacks, Witt rings
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