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 ${\mathrm{ed}}_k{X}_k\le {\mathrm{ed}}_K{X}_K$ if $X\to \mathrm{Spec}R$ is smooth and ${\mathrm{ed}}_k{X}_k\le {\mathrm{ed}}_K{X}_K+1$ in general.