Galois subspaces for compact Riemann surfaces of genus 2

1. Problem Statement

The paper addresses the classification and dimension calculation of Galois subspaces for compact Riemann surfaces of genus $g=2$.

• Context: Let $X$ be a compact Riemann surface of genus $g$, and $D$ a very ample divisor inducing an embedding $\phi_D: X \to \mathbb{P}^N$. A linear subspace $W \subset \mathbb{P}^N$ of codimension 2 is called a Galois subspace if the projection $\pi_W: \mathbb{P}^N \to \mathbb{P}^1$ composed with the embedding ($f_W = \pi_W \circ \phi_D$) yields a Galois cover.
• The Set $G_{X,D}$: The set of all such subspaces is denoted by $G_{X,D}$. Previous work by Auffarth and Rahausen established that for $g \geq 2$, this set decomposes into a disjoint union of projective spaces indexed by subgroups $H \leq \text{Aut}(X)$ such that the quotient $X/H \cong \mathbb{P}^1$.
• Specific Goal: While the decomposition structure is known, the dimensions of the constituent projective spaces were not fully calculated for genus 2. The paper aims to:
  1. Determine the dimension of these components when the divisor $D$ is induced by a subgroup of the automorphism group $\text{Aut}(X)$.
  2. Specifically characterize disjoint Galois subspaces (those where $W \cap \phi_D(X) = \emptyset$), which correspond to 0-dimensional components in the decomposition.

2. Methodology

The author employs algebraic geometry techniques specific to Riemann surfaces, focusing on divisor theory, linear equivalence, and the structure of automorphism groups.

• Decomposition Formula: The paper utilizes the isomorphism:
  $$G_{X,D} \cong \bigsqcup_{\substack{H \leq \text{Aut}(X) \\ X/H \cong \mathbb{P}^1}} X_{D, \pi_H}$$
  where $X_{D, \pi_H}$ is a projective space of dimension $\ell(D - D_H) - 1$. Here, $D_H$ is the divisor induced by the subgroup $H$ (the pullback of a point on $\mathbb{P}^1$), and $\ell(D - D_H) = \dim H^0(X, \mathcal{O}_X(D - D_H))$.
• Reduction to Subgroup Analysis: To study the components, the author reduces the problem to calculating $\ell(D_H - D_N)$ for subgroups $H, N \leq \text{Aut}(X)$ where $X/H \cong \mathbb{P}^1$ and $X/N \cong \mathbb{P}^1$.
• Key Lemmas on Linear Equivalence:
  • Lemma 1.1: Establishes that $D_H$ is independent of the choice of the point $p \in X$ up to linear equivalence.
  • Lemma 1.2 & 1.4: Provide conditions under which differences of induced divisors ($D_H - D_N$) are linearly equivalent to other induced divisors or the canonical divisor $K$. Specifically, if $|H| = |N|$, then $D_H \sim D_N$.
  • Corollary 1.3: For genus 2, if $H$ and $N$ both contain the hyperelliptic involution $\sigma$ and $|H| - |N| = 2$, then $D_H - D_N \sim K$ (the canonical divisor).
• Case-by-Case Classification: The author leverages the known classification of $\text{Aut}(X)$ for genus 2 surfaces (which are hyperelliptic). The possible groups are $C_2, C_2 \times C_2, D_4, C_{10}, D_6, C_3 \rtimes D_4, GL_2(\mathbb{F}_3)$.
• "Zig-Zag" Calculation: For each automorphism group, the author analyzes the subgroup lattice. By identifying intersections of subgroups and applying the lemmas regarding linear equivalence, they calculate $\ell(D_H - D_N)$ for all relevant pairs of subgroups.

3. Key Contributions

1. Theorem 0.2 (Main Result):
   • Part 1: For $G \not\cong C_{10}$, any two very ample divisors induced by subgroups of $G$ differ by a multiple of the canonical divisor $K$.
   • Part 2: In the set $G_{X, D_H}$, there is a one-to-one correspondence between disjoint Galois subspaces and subgroups of order $|H|$. This provides a precise count of disjoint Galois subspaces based on the subgroup structure.
2. Dimensional Decomposition: The paper provides explicit calculations for the dimensions of the projective space components of $G_{X, D_H}$ for all possible automorphism groups of genus 2 surfaces.
3. Generalization Limits: The author notes that while the lemmas used (1.2 and 1.4) hold for arbitrary genus $g \geq 2$, the specific structural results (Theorem 0.2) do not immediately generalize to $g > 2$ because the condition $X/H \cong \mathbb{P}^1$ is much more restrictive for higher genera.

4. Results

The paper culminates in a series of tables (Tables 1–5) detailing the dimensions of the components of $G_{X, D_H}$ for each possible automorphism group.

• Empty Sets: If $\text{Aut}(X) \cong C_2$ or $C_2 \times C_2$, there are no subgroups of order $\geq 5$, so $G_{X, D_H}$ is empty (since a very ample divisor on genus 2 requires degree $\geq 5$).
• Specific Dimensions:
  • For $D_4$ and $C_{10}$, the dimensions are calculated based on the difference in subgroup orders.
  • For $D_6$, $C_3 \rtimes D_4$, and $GL_2(\mathbb{F}_3)$, the author computes the dimensions for various subgroup orders $|H|$.
  • The dimension of a component corresponding to a subgroup $N$ is given by $\ell(D_H - D_N) - 1$.
  • The paper identifies that disjoint Galois subspaces (dimension 0) exist if and only if $\ell(D_H - D_N) = 1$, which occurs when $D_H \sim D_N$ (i.e., $|H|=|N|$).

Example Finding: For $G \cong GL_2(\mathbb{F}_3)$, the paper details how the intersection of subgroups (e.g., $C_8 \cap C_6 = Z(G)$) allows the reduction of the divisor difference to the canonical divisor, yielding specific dimension counts (e.g., $\ell = 2 \implies$ dimension 1).

5. Significance

• Completeness of Classification: This work fills a gap in the literature regarding Galois embeddings for genus 2 surfaces. While the general structure was known (Auffarth/Rahausen), the explicit dimensions and counts for the genus 2 case were previously unavailable.
• Connection to Automorphism Groups: The results tightly link the geometry of the embedding (Galois subspaces) to the group-theoretic structure of the surface's automorphisms.
• Disjoint Subspaces: The characterization of disjoint Galois subspaces is particularly significant as these are the "cleanest" projections where the branch locus behavior is most controlled. The one-to-one correspondence with subgroups of a specific order simplifies the search for such subspaces.
• Foundation for Higher Genus: By explicitly detailing the "Zig-Zag" method of using subgroup lattices and linear equivalence, the paper provides a potential framework for tackling similar problems in higher genera, despite the current limitations noted by the author.

In summary, the paper provides a complete algebraic description of the Galois subspace geometry for genus 2 Riemann surfaces, translating a geometric problem into a solvable group-theoretic calculation involving the automorphism groups of hyperelliptic curves.