On the Prym map of degree 4 cyclic covers of hyperelliptic curves

1. Problem Statement

The paper investigates the Prym map associated with degree 4 étale cyclic covers of hyperelliptic curves. Specifically, it focuses on the moduli space $RH_g[4]$, which parametrizes pairs $(H, \langle \eta \rangle)$ where $H$ is a hyperelliptic curve of genus $g$ and $\eta$ is a 4-torsion point in the Jacobian $J_H$ (with $\eta \notin J_H[2]$).

The author restricts the study to a specific irreducible component, denoted $RH_g[4]_{\text{hyp}}$, defined by the condition that the intermediate double cover $C' = \text{Spec}(\mathcal{O}_H \oplus \eta^2)$ is also hyperelliptic. The central questions addressed are:

1. Is the Prym map $P_g[4]: RH_g[4]_{\text{hyp}} \to \mathcal{A}_{(1,\dots,1,4,\dots,4)}$ injective for $g \geq 3$?
2. What is the geometric structure of the fibers of this map in the case $g=2$?

2. Methodology

The author employs a combination of algebraic geometry, the theory of abelian varieties, and explicit equation-based constructions. The methodology is structured as follows:

• Moduli Space Parametrization:
  The author establishes a bijection between the moduli space $RH_g[4]_{\text{hyp}}$ and a set of tuples of complex numbers, denoted $\Delta_g$. This set consists of unordered subsets $\{t_1, \dots, t_{2g-1}\} \subset \mathbb{C} \setminus \{0, 1\}$ modulo an equivalence relation involving inversion ($t_i \to 1/t_i$). This is achieved by analyzing the Weierstrass points of the hyperelliptic base curve $H$ and the line bundles defining the cover.

• Curve Equations and Quotients:
  Using the parametrization $\Delta_g$, the author explicitly writes down the defining equations for the hyperelliptic base curve $H$ and the various quotient curves arising from the deck transformations of the cover $C \to H$. Specifically, the curves $C_j$, $C_{j\sigma^2}$, and $C_{j\sigma, \sigma^2}$ (quotients of $C$ by involutions) are described as hyperelliptic curves with specific branch loci determined by the parameters $t_i$.

• Prym Variety Decomposition:
  A crucial step is the decomposition of the Prym variety $P(f)$. The author proves that for covers in $RH_g[4]_{\text{hyp}}$, the Prym variety is isogenous to the product of the Jacobians of three specific quotient curves:
  $$P(f) \sim JC_j \times JC_{j\sigma^2} \times JC_{j\sigma, \sigma^2}$$
  The isogeny $\mu$ is shown to be recoverable intrinsically from the polarized Prym variety $(P, \Xi)$.

• Fiber Analysis (Genus 2):
  For $g=2$, the quotient curves become elliptic curves. The author translates the condition $P_2[4](T) = P_2[4](S)$ into algebraic constraints on the parameters $T=\{t_1, t_2, t_3\}$ and $S=\{s_1, s_2, s_3\}$. This leads to the identification of fibers with intersections of quadrics in $\mathbb{P}^3$, which are shown to be elliptic curves.

• Injectivity Proof (Genus $g \geq 3$):
  The injectivity for higher genus relies on the Torelli Theorem. The author demonstrates that the Prym variety allows for the reconstruction of the quotient curves $C_j$ and $C_{j\sigma, \sigma^2}$. By comparing the sets of Weierstrass points of these reconstructed curves, the author proves that the original parameters $T$ are uniquely determined (up to the equivalence relation).

3. Key Contributions and Results

A. Parametrization of the Moduli Space

The paper provides a new, concrete description of the moduli space $RH_g[4]_{\text{hyp}}$ as the set $\Delta_g$. This reduces the study of these covers to the study of tuples of complex numbers, facilitating explicit calculations of equations for the curves involved.

B. Structure of Fibers for $g=2$

The author completely characterizes the fibers of the Prym map $P_2[4]$:

• Generic Fibers: For generic parameters, the non-empty fibers are isomorphic to the projective line $\mathbb{P}^1$ minus 8 points.
• Exceptional Fibers: There are two exceptional fibers (corresponding to specific values of the cross-ratios $\lambda_1, \lambda_2$) where the structure differs (involving glued points).
• Geometric Interpretation: The fibers are identified with the quotient of an elliptic normal curve (intersection of two quadrics in $\mathbb{P}^3$) by an involution, removing specific points where the involution is undefined or where the parameters violate the distinctness condition.

C. Injectivity for $g \geq 3$

The main theorem states that the Prym map $P_g[4]$ restricted to $RH_g[4]_{\text{hyp}}$ is injective for all $g \geq 3$.

• Mechanism: The proof relies on the fact that for $g \geq 3$, the system of equations relating the parameters of the base curve and the quotient curves becomes rigid. Specifically, if two sets of parameters yield isomorphic Prym varieties, the associated Möbius transformations must satisfy $u(x)^2 = v(x^2)$. For $g \geq 3$, the number of constraints forces $u(x) = x$ or $u(x) = 1/x$, implying the sets are equivalent in $\Delta_g$.

D. Decomposition of the Prym Variety

The paper establishes that for this specific component of the moduli space, the Prym variety decomposes into a product of Jacobians (up to isogeny). This contrasts with general cyclic covers where such a clean decomposition into Jacobians of lower-genus curves does not always exist.

4. Significance

• Extension of Known Results: While the injectivity of Prym maps for general cyclic covers has been studied (e.g., for $d \geq 6$ or specific prime powers), the case of degree 4 covers was less understood, particularly regarding the structure of fibers in low genus. This paper fills that gap for the hyperelliptic component.
• Explicit Geometry: The paper moves beyond abstract existence proofs by providing explicit equations for the curves and the moduli space. This allows for concrete computations and a deeper geometric understanding of the fibers.
• Novelty in Fiber Description: The author notes that while similar methods (using tuples of points) have been used for other Prym maps, this is the first paper to utilize these methods to describe positive-dimensional fibers explicitly.
• Limitations: The author acknowledges that the methods rely heavily on the hyperellipticity of the intermediate cover $C'$. Consequently, these techniques do not directly generalize to other irreducible components of $RH_g[4]$ where the intermediate curve is not hyperelliptic.

In summary, Shatsila's work provides a complete and explicit solution to the injectivity and fiber structure problems for degree 4 cyclic covers of hyperelliptic curves within the specific component where the intermediate cover remains hyperelliptic, bridging the gap between abstract moduli theory and explicit algebraic geometry.