Essential dimension and algebraic stacks

TL;DR

This paper introduces the concept of essential dimension for algebraic stacks, computes it for certain moduli stacks, and establishes new bounds and formulas for algebraic groups and related structures, with applications to quadratic forms.

Contribution

It defines and studies the essential dimension of algebraic stacks, providing explicit computations and new bounds for algebraic groups and finite p-groups, and applying results to quadratic form theory.

Findings

Computed essential dimension of Mgn and MgnBar stacks.

Established exponential lower bounds for spin groups.

Derived formulas for essential dimension of certain finite p-groups.

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.