The behavior of essential dimension under specialization

Abstract

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