Galois subspaces for smooth projective curves

TL;DR

This paper investigates the geometric structure of the set of linear subspaces in projective space that induce Galois morphisms from embedded smooth projective curves to the projective line, revealing their smooth projective variety structure.

Contribution

It characterizes the locus of such Galois-inducing subspaces as smooth projective varieties, detailing their structure for different genera of curves.

Findings

For genus g ≥ 2, the locus is a smooth projective variety with components isomorphic to projective spaces.

For genus 1, the locus is a smooth projective variety with components as projective bundles over étale quotients of the elliptic curve.

Explicit descriptions of these components are provided for the genus 1 case.

Abstract

Given an embedding of a smooth projective curve $X$ of genus $g\geq1$ into $\mathbb{P}^N$, we study the locus of linear subspaces of $\mathbb{P}^N$ of codimension 2 such that projection from said subspace, composed with the embedding, gives a Galois morphism $X\to\mathbb{P}^1$. For genus $g\geq2$ we prove that this locus is a smooth projective variety with components isomorphic to projective spaces. If $g=1$ and the embedding is given by a complete linear system, we prove that this locus is also a smooth projective variety whose positive-dimensional components are isomorphic to projective bundles over \'etale quotients of the elliptic curve, and we describe these components explicitly.