Essential dimension of moduli of curves and other algebraic stacks

TL;DR

This paper introduces a notion of essential dimension for algebraic stacks and determines the minimal parameters needed to define smooth or stable curves, and principally polarized abelian varieties, over a base field.

Contribution

It provides a complete answer for the essential dimension of moduli of smooth/stable curves and abelian varieties, extending the concept to algebraic stacks.

Findings

Calculated essential dimension for smooth/stable curves

Determined essential dimension for principally polarized abelian varieties

Connected essential dimension with moduli problems and algebraic stacks

Abstract

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).