Pseudoreflections on Prym Varieties

1. Problem Statement and Context

The paper investigates the existence of pseudoreflections of geometric origin acting on Prym varieties.

• Background: A previous result by Auffarth, Lahoz, and Naranjo ([ALA22]) established that for a Jacobian variety $J_C$ of a smooth curve $C$ of genus $g \geq 1$, if a finite group $G$ acts on $C$ such that the quotient $J_C/G$ is smooth, then $g \leq 3$. In other words, Jacobians of higher genus do not admit smooth quotients by groups containing pseudoreflections of geometric origin.
• The Question: Does a similar restriction hold for Prym varieties? Specifically, can a Prym variety $P(\tilde{C}, C)$ of arbitrarily large dimension possess a pseudoreflection of geometric origin such that the quotient is smooth?
• Definitions:
  • Prym Variety: Associated with an étale double cover $f: \tilde{C} \to C$ defined by a non-trivial 2-torsion point $\eta \in J_C[2]$. It is the connected component of the kernel of the norm map $Nm_f: J_{\tilde{C}} \to J_C$.
  • Geometric Origin: An automorphism on the Prym variety $P$ is of geometric origin if it is induced by an automorphism of the base curve $C$ that fixes $\eta$.
  • Pseudoreflection: An element $\sigma$ of a group acting on a manifold $X$ is a pseudoreflection at $x$ if $\sigma(x)=x$ and the fixed locus of $\sigma$ near $x$ has pure codimension 1. By the Chevalley-Shephard-Todd theorem, $X/G$ is smooth if and only if stabilizers are generated by pseudoreflections.

2. Methodology

The authors employ a combination of algebraic geometry tools, including:

• Riemann-Hurwitz Formula: To analyze the genera of quotient curves and the ramification of covering maps.
• Lefschetz Fixed Point Formula: To constrain the possible orders of automorphisms acting on the tangent spaces of the Prym variety.
• Moduli Space Analysis: Investigating the dimensions and irreducibility of loci within the moduli space of Prym varieties ($A_{g-1}$).
• Torelli Theorems for Prym Maps: Utilizing injectivity results for ramified Prym maps to reconstruct curves from their Prym varieties.
• Classification of Smooth Quotients: Applying results from [ALA20, ALA22] regarding the structure of groups acting on abelian varieties with smooth quotients.

The strategy involves:

1. Proving that pseudoreflections of geometric origin on Prym varieties must have order 2 or 4 (ruling out odd primes).
2. Analyzing the case of involutions (order 2) by decomposing the Prym variety via quotient curves.
3. Identifying three distinct irreducible families of Prym varieties that admit such pseudoreflections.
4. Proving these families are distinct and calculating their dimensions.
5. Restricting the possible groups $G$ that can act on these varieties.

3. Key Contributions and Results

A. Reduction to Orders 2 and 4

The authors first prove that any pseudoreflection of geometric origin on a Prym variety of dimension $g-1 \geq 4$ (i.e., $g \geq 5$) must have order 2 or 4.

• Proposition 3.3: Pseudoreflections of odd prime order $p \geq 3$ are impossible for $g \geq 3$.
• Proposition 3.4: If an automorphism of order 3 exists, the codimension of its fixed locus is at least 3, preventing it from being a pseudoreflection.
• Lemma 3.5: If an automorphism of order 4 acts as a pseudoreflection, its square (order 2) must also act as a pseudoreflection. Thus, the problem reduces to analyzing involutions.

B. Classification of Families

The main theorem (Theorem 1.1) states that for every $g \geq 5$, the locus of Prym varieties in $A_{g-1}$ possessing a pseudoreflection of geometric origin is the union of three distinct irreducible families:

1. Family 1 (Hyperelliptic Product Type):

   • Structure: $P \cong E \times J_{C'}$, where $E$ is an elliptic curve and $J_{C'}$ is a Jacobian of a hyperelliptic curve.
   • Dimension: $2g - 4$.
   • Description: Corresponds to the locus $A_1 \times JH_{g-2}$. The pseudoreflection acts as $-1$ on $E$ and identity on $J_{C'}$.

2. Family 2 (Bielliptic Type):

   • Structure: Prym varieties associated with bielliptic curves (curves admitting a degree 2 map to an elliptic curve).
   • Dimension: $2g - 3$.
   • Description: The image of the moduli space $RB_{g,2}$ under the Prym map.

3. Family 3 (Genus 2 Base Type):

   • Structure: Prym varieties arising from double covers of curves of genus 2, branched at $2g-6$ points.
   • Dimension: $2g - 3$.
   • Description: The image of the moduli space $RF_{g,3}$.

Distinctness: The authors prove (Section 4.5) that these families are distinct.

• Family 1 and 2 are disjoint (bielliptic curves of genus $\geq 4$ are not hyperelliptic).
• Family 1 and 3 are disjoint for $g \geq 6$; for $g=5$, their intersection has dimension at most 5, while the families have dimension 6 and 7 respectively.
• Family 2 and 3 have different closures (distinguished by the dimension of the singular locus of the theta divisor and the nature of the underlying curves).

C. Distinguished Example: Cubic Threefolds

In Section 4.4, the authors provide a geometric realization for the case $g=6$ (dimension 5).

• They show that Prym varieties with a pseudoreflection appear naturally as intermediate Jacobians of cubic threefolds with Eckardt points.
• An Eckardt point $p$ on a cubic threefold $X$ induces an involution $\sigma_p$. The intermediate Jacobian $J_X$ is isomorphic to the Prym variety of a specific double cover.
• This example demonstrates that a single Prym variety can possess multiple pseudoreflections of geometric origin (corresponding to different Eckardt points).

D. Restriction on Acting Groups

Section 5 restricts the possible finite groups $G$ of geometric origin that can act on a Prym variety with a smooth quotient.

• Proposition 5.1: If $g \geq 5$, $G$ must be a 2-group.
• If the representation is faithful and the polarization is irreducible, $G$ must be isomorphic to $(\mathbb{Z}/2\mathbb{Z})^i$ for $i \in \{1, 2, 3\}$.
• In the general case, $G$ is isomorphic to one of nine specific groups, including products of $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$, and semi-direct products involving $S_2$.

4. Significance

• Contrast with Jacobians: The paper establishes a fundamental difference between Jacobian varieties and Prym varieties. While Jacobians of genus $g > 3$ cannot admit smooth quotients by groups with pseudoreflections, Prym varieties of arbitrarily large dimension ($g \geq 5$) do admit such structures.
• Moduli Theory: It provides a complete classification of the loci in the moduli space of Prym varieties ($A_{g-1}$) where these special symmetries occur, identifying three explicit families.
• Geometric Applications: The connection to cubic threefolds with Eckardt points bridges the theory of Prym varieties with classical algebraic geometry of hypersurfaces, offering new examples of smooth quotients.
• Group Theory: It significantly narrows the search for groups acting on Prym varieties with smooth quotients, proving they must be 2-groups and listing the specific candidates.

In summary, the paper resolves the question of whether Prym varieties can have smooth quotients by geometric pseudoreflections, answering "yes" and providing a precise, three-part classification of where these phenomena occur in the moduli space.