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 1-dimensional noetherian local domain R with fraction field K and residue field k. We prove that ed Introduction, and the statement of the main theoremsLet k be a field, X → Spec k an algebraic stack, an extension of k, ξ ∈ X( ) an object of X over . If k ⊆ L ⊆ is an intermediate extension, we say, very naturally, 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

Stable pairs with a twist and gluing morphisms for moduli of surfaces

“…Assume that U ̸ = C. There is a map ϕ : U → M 1,1 . Up to replacing C by some root stack γ : C → C of C, with U ⊆ C the open locus over which γ is an isomorphism, we can extend ϕ to Φ : BPS22,BV23]). Consider the pull-back of the universal family via Φ, denoted by (X , S) → C, and let g ′ : (X ′ , S ′ ) → C be the corresponding map on coarse moduli spaces.…”

The behavior of essential dimension under specialization

“…(1) For a recent generalization of the inequality (1.4) in a different (stack-theoretic) direction, see [BV22]. Now suppose that Y is a generically free primitive G-variety defined over k. By a G-compression of Y we will mean a dominant G-equivariant rational map Y X, defined over k, where the G-action on X is again generically free and primitive.…”