Klein coverings over hyperelliptic genus 3 curves

1. Problem Statement

The paper investigates the injectivity of the Prym map for Klein coverings (étale Galois coverings with deck group $V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$) of smooth projective curves of genus 3.

Specifically, the authors focus on the case where the base curve $H$ is hyperelliptic. They aim to characterize the moduli space of such coverings and determine whether the Prym map, which associates a covering to its polarized Prym variety, is injective. While injectivity is known for étale double covers and cyclic coverings, the non-cyclic case (Klein coverings) presents unique challenges due to the structure of the automorphism group and the resulting decomposition of the Jacobian.

2. Mathematical Framework and Methodology

2.1. Definitions and Setup

• Klein Coverings: An étale covering $f: \tilde{C} \to H$ where the deck transformation group is isomorphic to $V_4 = \langle \sigma, \tau \rangle$.
• Construction: These coverings are constructed via a subgroup $\langle \eta, \xi \rangle \subset J_H[2]$ (the 2-torsion points of the Jacobian of $H$).
  • Bottom-up: Start with $H$ and a Klein subgroup of $J_H[2]$.
  • Top-down: Start with $\tilde{C}$ and a fixed-point-free $V_4 \subset \text{Aut}(\tilde{C})$.
• Prym Variety: $P(f)$ is the connected component of the kernel of the norm map $N_f: J_{\tilde{C}} \to J_H$. For genus 3 base curves, $\dim(P(f)) = 6$.
• Polarization: The restriction of the principal polarization of $J_{\tilde{C}}$ to $P(f)$ induces a non-principal polarization $\Xi$. The type of this polarization depends on whether the subgroup $\langle \eta, \xi \rangle$ is isotropic or non-isotropic with respect to the Weil pairing.

2.2. Classification of Cases

The authors classify Klein coverings of hyperelliptic genus 3 curves into four irreducible components based on the configuration of the generators $\eta, \xi$ in terms of Weierstrass points ($w_i$) and their isotropy:

1. Type I.1 (Non-isotropic): $\eta = w_1 - w_2$, $\xi = w_1 - w_3$. (Top curve $\tilde{C}$ is hyperelliptic).
2. Type I.2 (Non-isotropic): $\eta = w_1 - w_2$, $\xi = w_1 - w_3 + w_4 - w_5$.
3. Type II.1 (Isotropic): $\eta = w_1 - w_2 + w_3 - w_4$, $\xi = w_1 - w_2 + w_5 - w_6$.
4. Type II.2 (Isotropic): $\eta = w_1 - w_2$, $\xi = w_3 - w_4$.

2.3. Core Methodology: Reconstruction via Isotypical Decomposition

The central strategy to prove injectivity is reconstructing the covering curve $\tilde{C}$ from the polarized Prym variety $(P, \Xi)$. The proof relies on the following steps:

1. Automorphism Group Recovery:

   • The polarized Prym variety $(P, \Xi)$ determines a specific group of automorphisms $B \cong \mathbb{Z}_2^4$ acting on $P$.
   • This group distinguishes between genuine involutions (induced by automorphisms of $\tilde{C}$) and artificial involutions (induced by automorphisms of the Jacobian but not the curve).
   • Crucially, the presence of a hyperelliptic involution lift ($j$) in the hyperelliptic base case allows for a richer group structure ($B \cong \mathbb{Z}_2^4$) compared to the non-hyperelliptic case.

2. Isotypical Decomposition:

   • Using the action of $B$, the authors decompose the Prym variety $P$ into isotypical components (abelian subvarieties).
   • These components correspond to Jacobians of intermediate quotient curves (e.g., $C_\sigma, C_\tau, C_{j\tau}$, etc.).
   • The polarization type (e.g., $(1,1,1,2,2,4)$ vs $(1,1,1,1,4,4)$) and the dimension of fixed loci of involutions allow the authors to uniquely identify which component corresponds to which quotient curve.

3. Reconstruction via Fibered Products:

   • Once specific Jacobians (e.g., $J_{C_j}$ and $J_{C_{j\sigma}}$) are identified within $P$, the Strong Torelli Theorem is used to recover the curves $C_j$ and $C_{j\sigma}$.
   • The authors show that $\tilde{C}$ can be reconstructed as the normalized fibered product of these quotient curves over a common base (e.g., $C_j \times_E C_{j\sigma}$).
   • This reconstruction uniquely determines the original Klein subgroup and thus the covering.

3. Key Contributions and Results

3.1. Main Theorem (Injectivity on Hyperelliptic Base)

Theorem 0.1: The Prym maps for Klein coverings of hyperelliptic genus 3 curves are injective on each irreducible component of the moduli space.

• Isotropic Case: $P^{\text{iso}}_3: \mathcal{R}^{\text{iso}}_3 \to \mathcal{A}^{(1,1,1,2,2,4)}_6$ is injective.
• Non-Isotropic Case: $P^{\text{ni}}_3: \mathcal{R}^{\text{ni}}_3 \to \mathcal{A}^{(1,1,1,1,4,4)}_6$ is injective.
• The proof handles each of the four types (I.1, I.2, II.1, II.2) individually, utilizing the specific structure of the automorphism groups and fixed points for each case.

3.2. Moduli Space Structure

• The moduli space $\mathcal{R}^{V_4}_3$ of Klein coverings over hyperelliptic curves consists of four irreducible components.
• The authors provide explicit degrees for the forgetful map from these components to the moduli space of hyperelliptic curves $\mathcal{H}_3$:
  • Type I.1: Degree 56
  • Type I.2: Degree 280
  • Type II.1: Degree 105
  • Type II.2: Degree 210

3.3. Application to General Genus 3 Curves

Theorem 0.2 / Theorem 6.2: The Prym maps for Klein coverings over any genus 3 curve (not just hyperelliptic) are generically finite.

• The map is of degree 1 (injective) on the images of coverings of hyperelliptic curves of types I.2, II.1, and II.2.
• This suggests that the hyperelliptic locus serves as a "generic" test case for the injectivity of the map in the broader moduli space.

3.4. Geometric Insights

• Pseudoreflections: The paper identifies specific involutions in the Prym variety that act as pseudoreflections of geometric origin, linking the geometry of the covering to the structure of the Prym variety.
• Bielliptic Curves: In Type II.1, the authors utilize the fact that certain Prym varieties correspond to bielliptic genus 5 curves, which are not trigonal, to distinguish Jacobians from other abelian varieties.

4. Significance

1. Extension of Prym Injectivity: This work significantly extends the known results on the injectivity of the Prym map. While injectivity was established for double covers and cyclic covers, this paper resolves the case for non-cyclic (Klein) coverings, a more complex scenario involving non-abelian deck groups.
2. Moduli Space Characterization: It provides a complete characterization of the moduli space of Klein coverings over hyperelliptic curves, identifying the irreducible components and their geometric properties.
3. Reconstruction Techniques: The methodology of using the isotypical decomposition of the Prym variety combined with polarization types to reconstruct the underlying curve offers a powerful tool for studying other non-cyclic coverings.
4. Generic Finiteness: The result that the map is generically finite for all genus 3 curves (Theorem 6.2) is a major step toward the conjecture that the Prym map is generically injective for Klein coverings of any genus 3 curve, answering a question posed in the literature.

In summary, the paper establishes that the geometric data of a Klein covering of a hyperelliptic genus 3 curve is uniquely encoded in its polarized Prym variety, providing a robust framework for understanding the geometry of these higher-genus coverings.