Essential dimension and algebraic stacks

2007/01/31 by Patrick Brosnan, Zinovy Reichstein, Brosnan, Patrick +3
Mathematics · #11E04 #14A20 #14H10 #20G15 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:11E04 #msc:14A20 #msc:14H10 #msc:20G15

paper · pdf · doi:10.48550/arxiv.math/0701903

52 pages

arxiv created 2007/01/31 · arxiv updated 2009/12/01

Abstract

We define and study the essential dimension of an algebraic stack. We compute the essential dimension of the stacks Mgn and MgnBar 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.

Related

• Essential p-dimension of algebraic tori
• Essential dimension of the spin groups in characteristic 2
• Essential dimensions of finite groups
• On the essential dimension of unipotent algebraic groups
• Essential dimension of abelian varieties over number fields