Galois subspaces for compact Riemann surfaces of genus 2

TL;DR

This paper investigates the structure of Galois subspaces associated with genus 2 Riemann surfaces, focusing on the dimensions of certain projective spaces linked to automorphism subgroups.

Contribution

It computes the dimensions of projective spaces in the decomposition of Galois subspaces for genus 2 surfaces when divisors are induced by automorphism subgroups.

Findings

Determines the dimension of projective spaces in the decomposition of G_{X,D}

Connects the structure of Galois subspaces to automorphism subgroups

Provides explicit calculations for genus 2 cases

Abstract

Let $X$ be a compact Riemann surface of genus 2 and $D$ a very ample divisor with $\phi_D$ its associated embedding into $\mathbb{P}^{n}$. We consider the set $G_{X,D}$ of linear subspaces $W$ of $\mathbb{P}^n$ of codimension $2$ with projection $\pi_W$ such that $f_W = \pi_W \circ \phi_D$ is Galois, i.e. $f_W^*k(\mathbb{P}^1) \subseteq k(X)$ is a Galois extension. It is known that $G_{X,D}$ is isomorphic to a disjoint union of projective spaces. In this article, we calculate the dimension of projective spaces in the decomposition of $G_{X,D}$, when $D$ is induced by a subgroup of $\mathrm{Aut}(X)$.