The behavior of essential dimension under specialization

Abstract

Let A be a discrete valuation ring with generic point η and closed point s. We show that in a family of torsors over Spec(A), the essential dimension of the torsor above s is less than or equal to the essential dimension of the torsor above η. We give two applications of this result, one in mixed characteristic, the other in equal characteristic.

The genericity theorem for the essential dimension of tame stacks

“…Theorem 1.2 also implies a slightly weaker version of Theorem 1.1, in which the morphism X → Spec k is assumed to be smooth. Theorems 1.2 and 1.3 are related in spirit with the results in [16,17].…”