Galois subspaces for compact Riemann surfaces of genus 2

Abstract

Let $X$ be a compact Riemann surface of genus 2 and $D$ a very ample divisor with $φ_D$ 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 $π_W$ such that $f_W = π_W \circ φ_D$ 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 $\mathrm{Aut}(X)$.

Figures (1)

• Figure 4: Dimension components of $G_{X,D}$ with $\text{Aut}(X) \cong C_3 \rtimes D_4$

Theorems & Definitions (17)

• Definition 1
• Theorem 2
• Remark 1
• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Corollary 1.3
• proof
• Lemma 1.4