A New Curve Algebraically but not Rationally Uniformized by Radicals

TL;DR

This paper constructs a genus 9 algebraic curve that cannot be rationally uniformized by radicals, but can be algebraically uniformized via a related curve, providing a novel example in algebraic geometry.

Contribution

It introduces a new example of a curve with algebraic but not rational uniformization by radicals, expanding understanding of such curves.

Findings

Curve C of genus 9 constructed in the second symmetric product of a genus 2 curve.

C has no map onto P^1 with solvable Galois group.

Existence of a curve C' mapping onto C with a finite morphism to P^1 with solvable Galois group.

Abstract

We give a new example of a curve C algebraically, but not rationally, uniformized by radicals. This means that C has no map onto the projective line P^1 with solvable Galois group, while there exists a curve C' that maps onto C and has a finite morphism to P^1 with solvable Galois group. We construct such a curve C of genus 9 in the second symmetric product of a general curve of genus 2. It is also an example of a genus 9 curve that does not satisfy condition S(4,2,9) of Abramovich and Harris.