The genericity theorem for the essential dimension of tame stacks

Abstract

Let X be a regular tame stack. If X is locally of finite type over a field, we prove that the essential dimension of X is equal to its generic essential dimension, this generalizes a previous result of P. Brosnan, Z. Reichstein and the second author. Now suppose that X is locally of finite type over a 1dimensional local noetherian domain R with fraction field K and residue field k. We prove that ed Introduction, and the statement of the main theoremLet k be a field, X → Spec k an algebraic stack, ℓ an extension of k, ξ ∈ X(ℓ) an object of X over ℓ. If k ⊆ L ⊆ ℓ is an intermediate extension, we say, very naturally, that that L is a field of definition of ξ if ξ descends to L. The essential dimension ed k ξ, which is either a natural number or +∞, is the minimal transcendence degree of a field of definition of ξ. If X is of finite type then ed k ξ is always finite.This number ed k ξ is a very natural invariant, which measures, essentially, the number of independent parameters that are needed for defining ξ. The essential dimension ed k X of X is the supremum of the essential dimension of all objects over all extensions of k (if X is empty then ed k X is −∞). This number is the answer to the question "how many independent parameters are needed to define the most complicated object of X?". For example, this is a very natural question for the stack M g of smooth projective curves of genus g.Essential dimension was introduced for classifying stacks of finite groups in [BR97], with a rather more geometric language. Since then it has been actively investigated by many mathematicians. It has been studied for classifying stacks of positive dimensional algebraic group starting from [Rei00], and for more general classes of geometric and algebraic objects in [BRV07]. See [BF03,BRV11,Rei10,Mer13] for an overview of the subject.Suppose that X is an integral algebraic stack which is locally of finite type over k. We can define the generic essential dimension ged k X (see [BRV11, Definition 3.3]) as the supremum of all ed k ξ taken over all ξ ∈ X(ℓ) such that the associated morphism ξ : Spec ℓ → X is dominant. For example, if X has finite inertia and X → M is its moduli space, then M is an integral scheme over K; if k(M ) is its field of rational functions, the pullback X k(M) → Spec k(M ) is a gerbe (the generic