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Essential dimension relative to branched covers of degree at most n

Benson Farb, Jesse Wolfson

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.

Essential dimension relative to branched covers of degree at most n

Abstract

We prove for various finite groups and integers that there are families of equations with Galois group that cannot be simplified to a one-parameter family even after adjoining a root of a polynomial of degree at most . In more geometric language, there are -varieties with the following property: for any -equivariant branched cover of degree , there is no dominant rational -map to any -curve . The method of proof is new, and applies in cases where previous methods do not.
Paper Structure (4 sections, 8 theorems, 22 equations)

This paper contains 4 sections, 8 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Rational functions on curves
  3. Induced Actions on Unions of Rational Curves
  4. Finishing the proofs of Theorem \ref{['t:main']} and Corollary \ref{['cor:main5']}

Key Result

Theorem 1.3

Let $k$ be a field of characteristic 0 with $\sqrt{5}\in k$. Let $X$ be any $A_5$-variety over $k$. Then $X$ has an $A_5$-equivariant branched coverBy a degree n branched cover we mean a generically $n$-to-$1$, dominant rational map $\tilde{X}\dashrightarrow X$.$\widetilde{X}\dashrightarrow X$ of d

Theorems & Definitions (17)

  • Example 1.1: Kummer's theorem
  • Definition 1.2: Essential dimension
  • Theorem 1.3: Klein's Normalformsatz
  • Remark 1.5
  • Theorem 1.6: Main Theorem
  • Corollary 1.7: Sample results
  • Theorem 2.1: Castelnuovo's Inequality
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • ...and 7 more