The behavior of essential dimension under specialization II

TL;DR

This paper investigates how the essential dimension of algebraic group actions varies in families, showing it can decrease on certain subsets but remains stable elsewhere, with multiple applications.

Contribution

It establishes that the essential dimension can drop on a countable union of Zariski closed subsets in a family of algebraic varieties, under mild conditions.

Findings

Essential dimension may decrease on countable Zariski closed subsets.

Essential dimension remains constant outside these subsets.

Applications demonstrate the practical implications of this behavior.

Abstract

Let $G$ be a linear algebraic group over a field. We show that, under mild assumptions, in a family of primitive generically free $G$-varieties over a base variety $B$ the essential dimension of the geometric fibers may drop on a countable union of Zariski closed subsets of $B$ and stays constant away from this countable union. We give several applications of this result.