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A note on smooth quotients of Prym varieties

This paper establishes that for base curves of genus $g \geq 4$ , pseudoreflections of geometric origin on Prym varieties of étale double covers have order 2, and consequently, for $g \geq 5$ , any non-trivial finite group of such automorphisms yielding a smooth quotient must be isomorphic to $\mathbb{Z}/2\mathbb{Z}$ or $(\mathbb{Z}/2\mathbb{Z})^2$ (the latter only occurring for $g \leq 7$ ), thereby sharpening previous results by Auffarth, Lahoz, and Naranjo.

Original authors: Anatoli Shatsila

Published 2026-08-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anatoli Shatsila

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there is a field dedicated to understanding shapes that exist in many dimensions, far beyond the three we experience in daily life. Among these complex shapes are objects called abelian varieties, which can be thought of as highly structured, multi-dimensional doughnuts. They are not just random shapes; they carry a specific kind of internal geometry, like a grid of lines woven into their fabric, which mathematicians call a polarization. These objects are central to number theory and the study of curves, acting as a bridge between the geometry of shapes and the arithmetic of numbers. Within this world, there is a special family of these multi-dimensional doughnuts known as Prym varieties. They arise when you take a smooth curve and create a double cover of it, essentially wrapping a new curve around the old one twice without any overlaps or tears. The Prym variety is the part of this new structure that captures the unique difference between the two layers. Mathematicians are deeply interested in how these shapes can be transformed or rotated by symmetries, and what happens to the shape when you fold it along those symmetries. If the folding process is too rough, the resulting shape develops sharp corners or singularities, much like crumpling a piece of paper. However, if the folding is done with perfect precision, the result is a smooth, clean shape. Understanding which symmetries allow for this smooth result is a key question in the field, as it reveals the fundamental limits and possibilities of these geometric forms.

A recent study by Anatoli Shatsila tackles this question by investigating a specific type of symmetry on Prym varieties. The researcher focused on symmetries that come directly from the geometry of the underlying curves, rather than abstract mathematical constructions. These are called symmetries of geometric origin. The study asks a precise question: if you have a Prym variety coming from a curve with a certain number of holes, known as its genus, what kinds of symmetries can you apply to it so that the folded result remains perfectly smooth? The paper specifically looks at a type of symmetry called a pseudoreflection. In simple terms, a pseudoreflection is a transformation that leaves almost everything in place, fixing a large flat slice of the space, while only twisting a single direction. The author proves that for curves with a genus of four or higher, any such symmetry of geometric origin must be a simple flip, turning the space over exactly once. This finding rules out the possibility of more complex twists, such as those that would require four steps to return to the starting position. Before this work, it was known that these symmetries could be simple flips or more complex quarter-turns, but the new research demonstrates that the quarter-turns are impossible in this specific context when the curve is sufficiently complex.

Building on this discovery, the paper explores what happens when a whole group of these symmetries acts together on the Prym variety. The goal is to find all possible groups of symmetries that can fold the shape smoothly without creating any rough edges. The author shows that for curves with a genus of five or higher, the only groups that can do this are very small and simple. Specifically, the group can only consist of a single flip, or a combination of two independent flips. Any larger or more complex group of symmetries, including those that might involve quarter-turns or more intricate patterns, is proven to be impossible if the resulting shape is to remain smooth. The study goes further to establish a strict limit on the size of the curve for the two-flip scenario to work. It turns out that this specific case can only happen if the curve has a genus of seven or less. If the curve has more than seven holes, the only possible smooth symmetry group is the single flip. This narrows down the possibilities significantly, eliminating several complex groups that previous researchers had listed as potential candidates.

The paper also addresses the certainty of these findings. The author provides rigorous proofs that leave no room for doubt regarding the elimination of the more complex symmetries. For the case of the two-flip group, the work demonstrates that it is not just a theoretical possibility but a real occurrence, provided the curve is not too large. The research confirms that the earlier lists of possible groups were too broad and needed to be trimmed. By carefully analyzing the branching points where the curves wrap around each other and how the symmetries interact with the geometry of the space, the author constructs a logical argument that holds up under scrutiny. The result is a much clearer picture of the landscape of these geometric objects. It tells us that nature, in the form of these mathematical shapes, is more restrictive than previously thought. The symmetries that preserve smoothness are rare and simple, confined to a small set of possibilities that depend directly on the complexity of the underlying curve. This work refines our understanding of the relationship between the geometry of curves and the structure of the abelian varieties they produce, offering a definitive answer to a question that had remained open for some time.

Technical Summary: A Note on Smooth Quotients of Prym Varieties

Problem Statement
This paper investigates the structure of automorphisms of geometric origin acting on Prym varieties $P(\tilde{C}/C)$ associated with étale double covers $f: \tilde{C} \to C$ of a smooth projective curve $C$ of genus $g$ . Specifically, the study focuses on pseudoreflections—automorphisms whose differential on the tangent space fixes a hyperplane pointwise—and the conditions under which a finite group $G$ of such automorphisms acts faithfully on $P$ such that the quotient $P/G$ is smooth.

Previous work by Auffarth, Lahoz, and Naranjo (cited as [3]) established that for $g \ge 5$ , the locus of Prym varieties admitting such pseudoreflections consists of three explicit families. They determined that the order of a geometric pseudoreflection is either 2 or 4 and provided a list of nine possible groups $G$ that could act faithfully with a smooth quotient. However, it remained unknown whether all nine groups could actually occur with smooth quotients, and whether the order-4 possibility could be eliminated for higher genera.

Methodology
The author employs a combination of representation theory, the geometry of abelian varieties, and the Riemann-Hurwitz formula applied to intermediate covers. The core methodology involves:

Lifting Automorphisms: Analyzing the lift of a pseudoreflection $\alpha \in \text{Aut}(P, \Xi)$ to an automorphism $\sigma$ of the covering curve $\tilde{C}$ that commutes with the covering involution $\iota$ .
Ramification Analysis: Utilizing the properties of pseudoreflections (specifically their eigenvalues on the space of holomorphic differentials) to compute the genera of intermediate quotient curves (e.g., $\tilde{C}/\langle \sigma \rangle$ , $\tilde{C}/\langle \iota\sigma \rangle$ ).
Contradiction via Riemann-Hurwitz: Deriving contradictions for specific group structures by comparing the number of branch points required by the ramification data against the constraints imposed by the Riemann-Hurwitz formula.
Decomposition of Pryms: For the classification of groups, the paper utilizes the Chevalley-Shephard-Todd theorem (which implies that a smooth quotient requires the group to be generated by pseudoreflections) and the classification of smooth quotients of principally polarized abelian varieties by Auffarth, Lahoz, and Naranjo ([2]).

Key Contributions and Results

Elimination of Order 4 (Theorem 1.1): The primary result proves that for genus $g \ge 4$ , every pseudoreflection of geometric origin on a Prym variety has order 2. This eliminates the order-4 case previously considered possible. The proof involves a detailed analysis of the ramification data of intermediate covers. The author demonstrates that assuming an order-4 pseudoreflection leads to a contradiction in the number of branch points for $g \ge 5$ , and a separate monodromy argument rules out the case for $g=4$ . The bound is shown to be sharp via an explicit construction for $g=3$ (Example 3.2).
Classification of Smooth Quotients (Theorem 1.2): Building on the elimination of order-4 elements and the list of nine candidate groups from [3], the paper classifies the possible groups $G$ acting faithfully on $P$ with a smooth quotient $P/G$ for $g \ge 5$ .
- The only possible non-trivial groups are $\mathbb{Z}/2\mathbb{Z}$ and $(\mathbb{Z}/2\mathbb{Z})^2$ .
- Groups containing elements of order 4 (such as $\mathbb{Z}/4\mathbb{Z}$ , $(\mathbb{Z}/4\mathbb{Z})^2$ , etc.) are eliminated because they cannot be generated solely by involutions (pseudoreflections of order 2).
- Dihedral groups and $(\mathbb{Z}/2\mathbb{Z})^3$ are eliminated through arguments involving the decomposition of the Prym variety and the irreducibility of theta divisors.
- Furthermore, the paper establishes that if $G \cong (\mathbb{Z}/2\mathbb{Z})^2$ , then the genus must satisfy $g \le 7$ . For $g > 7$ , only $G \cong \mathbb{Z}/2\mathbb{Z}$ is possible.

Significance
The paper sharpens the results of Auffarth, Lahoz, and Naranjo by significantly narrowing the landscape of possible automorphism groups for smooth quotients of Prym varieties. By proving that pseudoreflections of geometric origin must be involutions for $g \ge 4$ , the author reduces the list of potential symmetry groups from nine to two. This provides a more precise understanding of the geometry of Prym varieties with smooth quotients, confirming that higher-order symmetries and larger elementary abelian 2-groups are obstructed by the interplay between the curve's genus and the ramification of the covering maps.

Author's Note on Development
The author acknowledges that the statement of Theorem 1.1 and the main ideas behind its proof were suggested by an LLM (ChatGPT 5.6 Sol), which served as the starting point for the work. The full text and the rigorous arguments presented are the responsibility of the author.

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