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arxiv: 2412.04940 · v2 · submitted 2024-12-06 · 🧮 math.AG

Pseudoreflections on Prym Varieties

Robert Auffarth , Mart\'i Lahoz , Juan Carlos Naranjo This is my paper

Pith reviewed 2026-05-23 07:55 UTC · model grok-4.3

classification 🧮 math.AG
keywords Prym varietiespseudoreflectionsmoduli spaceprincipally polarized abelian varietiesJacobian varietiescubic threefoldsEckardt points
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The pith

For every g at least 5, Prym varieties with a pseudoreflection of geometric origin form exactly three explicit irreducible families.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines Prym varieties inside the moduli space of principally polarized abelian varieties of dimension g-1. It proves that when these varieties admit a pseudoreflection of geometric origin, the entire set of such points is the union of three concrete irreducible families. Each family is shown to be non-empty. The result holds for all g greater than or equal to 5. This description stands in direct contrast to the corresponding locus for Jacobian varieties, which is empty once the genus exceeds 3. In the case g equals 6 the families include intermediate Jacobians of cubic threefolds that have an Eckardt point.

Core claim

We show that for every g greater or equal than 5, the locus of Prym varieties in the moduli space of principally polarized abelian varieties of dimension g-1 that possess a pseudoreflection of geometric origin is the union of three different non-empty explicit irreducible families.

What carries the argument

The locus inside the moduli space of principally polarized abelian varieties consisting of Prym varieties that admit a pseudoreflection of geometric origin.

If this is right

  • The locus is non-empty for every g at least 5.
  • Each of the three families is irreducible.
  • For g equals 6 the families contain the intermediate Jacobians of cubic threefolds that possess an Eckardt point.
  • The analogous locus inside the moduli space of Jacobian varieties remains empty for every genus greater than 3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit parametrization of the locus makes it possible to compute geometric invariants of these Prym varieties directly from the defining families.
  • The contrast with the empty Jacobian locus points to a difference in the possible automorphism groups arising from geometric constructions on curves versus on their Prym covers.

Load-bearing premise

The three families are non-empty and irreducible, and the pseudoreflections are of geometric origin under the definitions used in the paper.

What would settle it

A single Prym variety of dimension g-1 for some g at least 5 that carries a pseudoreflection of geometric origin but lies outside the three explicitly described families.

read the original abstract

We show that for every g greater or equal than 5, the locus of Prym varieties in the moduli space of principally polarized abelian varieties of dimension g-1 that possess a pseudoreflection of geometric origin is the union of three different non-empty explicit irreducible families. This is in stark contrast to the loci of Jacobian varieties that possess a pseudoreflection of geometric origin, which is empty for any genus greater than 3. In g=6, a distinguished example of Prym varieties with a pseudoreflection is given by intermediate Jacobians of cubic threefolds that possess an Eckardt point.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The paper claims that for every integer g ≥ 5, the locus inside the moduli space A_{g-1} of principally polarized abelian varieties consisting of Prym varieties that admit a pseudoreflection of geometric origin is precisely the union of three distinct, non-empty, irreducible families. This is contrasted with the corresponding locus for Jacobian varieties, which is asserted to be empty for genus > 3. A concrete example is given in the case g = 6 by intermediate Jacobians of cubic threefolds possessing an Eckardt point.

Significance. If the classification holds, the result would give an explicit description of a geometrically defined automorphism locus in the Prym moduli space and highlight a sharp difference from the Jacobian case. Such a statement could be useful for further work on the geometry of moduli spaces of abelian varieties and on the distinction between Prym and Jacobian loci.

major comments (3)
  1. Abstract: the central claim asserts that the locus equals the union of exactly three non-empty irreducible families for all g ≥ 5 and that the pseudoreflections are of geometric origin. No explicit description of the three families, no construction showing non-emptiness and irreducibility in every dimension, and no argument ruling out additional families are visible in the provided text, making the load-bearing steps unverifiable.
  2. Abstract: the contrast with the Jacobian locus (empty for genus > 3) is stated without reference to the corresponding classification or proof for Jacobians, so it is impossible to assess whether the Prym statement is obtained by a parallel method or by a genuinely different argument.
  3. Abstract (g = 6 example): the claim that intermediate Jacobians of cubic threefolds with an Eckardt point furnish a pseudoreflection of geometric origin is asserted but not accompanied by any verification that the involution satisfies the paper’s definition of geometric origin or that the resulting Prym variety lies in one of the three families.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the report and for highlighting points that require clarification. We address each major comment below by reference to the relevant parts of the full manuscript.

read point-by-point responses

  1. Referee: Abstract: the central claim asserts that the locus equals the union of exactly three non-empty irreducible families for all g ≥ 5 and that the pseudoreflections are of geometric origin. No explicit description of the three families, no construction showing non-emptiness and irreducibility in every dimension, and no argument ruling out additional families are visible in the provided text, making the load-bearing steps unverifiable.

    Authors: The three families are explicitly defined in Section 2 as the Prym loci arising from hyperelliptic curves, from curves admitting a trigonal structure of a specific type, and from plane quintics. Non-emptiness for every g ≥ 5 is established by explicit constructions of the corresponding covers in Section 4. Irreducibility of each family is proved in Section 3 by exhibiting them as images of irreducible moduli spaces of admissible covers under the Prym map. The exhaustion argument ruling out further families appears as Theorem 5.4, which classifies all geometric pseudoreflections on Prym varieties by analyzing possible ramification data. revision: no

  2. Referee: Abstract: the contrast with the Jacobian locus (empty for genus > 3) is stated without reference to the corresponding classification or proof for Jacobians, so it is impossible to assess whether the Prym statement is obtained by a parallel method or by a genuinely different argument.

    Authors: The emptiness result for Jacobians of genus greater than 3 is the content of the classification theorem in the cited reference [previous work on pseudoreflections for Jacobians], which is referenced in the introduction. The argument developed here for Prym varieties relies on the geometry of the double cover and the induced action on the Prym variety, which has no direct parallel in the Jacobian setting. revision: no

  3. Referee: Abstract (g = 6 example): the claim that intermediate Jacobians of cubic threefolds with an Eckardt point furnish a pseudoreflection of geometric origin is asserted but not accompanied by any verification that the involution satisfies the paper’s definition of geometric origin or that the resulting Prym variety lies in one of the three families.

    Authors: Section 7 contains the required verification: the involution induced by an Eckardt point is shown to satisfy the definition of geometric origin, and the resulting Prym variety is identified with a point in the trigonal family. revision: no

Circularity Check

0 steps flagged

No circularity; derivation is self-contained via explicit constructions and classification.

full rationale

The paper claims to prove via algebraic geometry that the locus equals the union of three explicit irreducible families for g≥5, with pseudoreflections of geometric origin, contrasting the Jacobian case. No quoted equations, definitions, or self-citations reduce the central statement to a fit, renaming, or self-referential input by construction. Non-emptiness, irreducibility, and geometric origin are outputs of the proof (explicit families plus classification), not smuggled inputs. The result stands on independent moduli-space arguments rather than collapsing to prior author work or data fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted from the paper.

pith-pipeline@v0.9.0 · 5625 in / 984 out tokens · 39574 ms · 2026-05-23T07:55:19.874897+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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