Essential dimension relative to branched covers of degree at most n

TL;DR

This paper demonstrates that for certain finite groups and degrees, there are Galois covers that cannot be simplified to one-parameter families, revealing new geometric obstructions in the theory of branched covers.

Contribution

It introduces a novel method to prove the non-existence of simplified Galois covers for specific groups and degrees, expanding the understanding of essential dimension in algebraic geometry.

Findings

Existence of Galois covers that cannot be reduced to one-parameter families.

New proof technique applicable where previous methods fail.

Identification of geometric obstructions in branched covers.

Abstract

We prove for various finite groups $G$ and integers $n\geq 1$ that there are families of equations with Galois group $G$ that cannot be simplified to a one-parameter family even after adjoining a root of a polynomial of degree at most $n$. In more geometric language, there are $G$-varieties $X$ with the following property: for any $G$-equivariant branched cover $\widetilde{X}\to X$ of degree $\leq n$, there is no dominant rational $G$-map $\widetilde{X}\dashrightarrow C$ to any $G$-curve $C$. The method of proof is new, and applies in cases where previous methods do not.