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arxiv: 2605.21675 · v2 · pith:ABIPS4ZBnew · submitted 2026-05-20 · 🧮 math.AG

Non--tautological cycles on Prym moduli spaces

Bogdan Carasca , Riccardo Redigolo This is my paper

Pith reviewed 2026-06-30 16:50 UTC · model grok-4.3

classification 🧮 math.AG
keywords Prym moduli spacesChow ringsnon-tautological cyclesbi-elliptic Prym curvesPrym curvestautological ringmoduli of curves
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The pith

The fundamental class of the bi-elliptic Prym locus component in genus 8 is not tautological in the Chow ring of the Prym moduli space.

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

The paper shows that the Chow ring of the moduli space of Prym curves R_g contains classes outside its tautological subring starting at genus 8. It exhibits the fundamental class of the component RB_8^0 of the bi-elliptic Prym locus as one such non-tautological cycle. This improves the known bound for non-tautology relative to the moduli space of curves. The same argument establishes non-tautology for the compact pointed spaces ar R_{g;2m} whenever g + m is at least 8.

Core claim

The authors prove the non-tautology of the class [RB_8^0] in CH^*(R_8). The locus RB_g^0 parametrizes covers [ ilde C / C] such that if C o E is the bi-elliptic structure on C, the composition ilde C o E factors through an elliptic cover of E. The same non-tautology is shown for the compact moduli spaces ar R_{g;2m} when g + m \geq 8.

What carries the argument

The component RB_g^0 of the bi-elliptic Prym locus, whose fundamental class is shown to lie outside the tautological subring of the Chow ring of R_g.

If this is right

  • The Chow ring of R_8 properly contains its tautological subring.
  • Non-tautological classes exist in CH^*(ar R_{g;2m}) for all g + m \geq 8.
  • The bi-elliptic Prym locus supplies a concrete cycle witnessing non-tautology at genus 8.
  • The same construction yields non-tautological classes in the pointed compact Prym spaces at the threshold g + m = 8.

Where Pith is reading between the lines

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

  • The method of factoring covers through elliptic curves may produce non-tautological classes in other double-cover moduli problems at low genus.
  • The result suggests that the Chow ring of Prym moduli spaces is strictly larger than that of ordinary curve moduli at the same genus.
  • Similar bi-elliptic or multi-elliptic loci could be tested for non-tautology in related spaces such as moduli of abelian varieties or covers with higher degree.

Load-bearing premise

The component RB_g^0 is a well-defined closed subvariety of R_g whose fundamental class can be compared directly to the tautological subring.

What would settle it

An explicit computation in the Chow ring of R_8 showing that [RB_8^0] equals a linear combination of tautological classes, or a degeneration argument proving that the class vanishes or lies in the tautological ring.

Figures

Figures reproduced from arXiv: 2605.21675 by Bogdan Carasca, Riccardo Redigolo.

Figure 1
Figure 1. Figure 1: The Harmonic Morphisms giving rise to the gluing maps in (1) and (2) (4) Define the clutching maps R ′ g−1,2r;m+2 → R ′ g,2r;m, [C/C, e (pi) 2r i=1; (xj ) m j=1,(x ± j ) m j=1] 7→ [C/e {x ± m+1 ∼ x ± m+2} → C/{xm+1 ∼ xm+2},(pi) 2r i=1; (xj ) m j=1,(x ± j ) m j=1]. i g−i x− 2g−2i−1+r x x + x− 2r x i x + 2r ∗ ∗ + ∗− ϕ3: i g−1 m− 2g−3+r m m+ 2r 2r ∗ ∗ + ∗− ϕ4 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Harmonic Morphisms giving rise to the gluing maps in (3) and (4) (5) Define the clutching map Mg−1,m+2 → R ′ g;m, [C,(xj ) m+2 j=1 ] 7→ [(C ∪{xm+1,xm+2} C)/C,(xj ) m j=1,(x ± j ) m j=1]. See [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Harmonic Morphisms giving rise to the clutching maps in (5) and (6) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The image of [E, E e ] under θ As in [vZ18, Proof of Proposition 5], we are left to show that ζ : R′ 1;7 → F is surjective on geometric points. By definition, a point in F(C) is given by a triple (Ee1/E1, Ee2/E2), Se → S → T, γ where (Ee1/E1, Ee2/E2) ∈ (R′ 1;7 × R′ 1;7)(C), (Se → S → T) ∈ RAdm(g, 1)0 2m, and γ is an isomorphism between χ(Ee1/E1, Ee2/E2) and ϕ(Se → S → T). Let (E, E e ) := χ(Ee1/E1, Ee2/E2… view at source ↗
read the original abstract

We denote by $\mathcal{R}_{g;m}$ the moduli space of $m$--pointed Prym curves of genus $g$, that is, tuples $[\widetilde C / C; x_1, \dots, x_m]$ where $[C, x_1, \dots, x_m]$ is an $m$--pointed curve of genus $g$ and $\widetilde C/ C$ is an \'etale double cover of $C$. In this paper, we address the problem of the non--tautology of the Chow ring of $\mathcal{R}_{g;m}$. The locus which allows us to achieve earlier bounds for the non--tautology of $\mathrm{CH}^\bullet(\mathcal{R}_{g})$ compared to $\mathcal{M}_g$ is the component $\mathcal{R}\mathcal{B}_g^0$ of the locus of bi--elliptic Prym curves. This parametrises covers $[\widetilde C/ C]$ such that, if $C \rightarrow E$ is the bi--elliptic structure, the composition $\widetilde C \rightarrow E$ factors through an elliptic cover of $E$. Our main contribution is thus the non--tautology of the class $[\mathcal{R}\mathcal{B}_8^0] \in \mathrm{CH}^*(\mathcal{R}_8)$. In the course of establishing this theorem, a similar result for the compact moduli spaces $\overline{\mathcal{R}}_{g; 2m}$ for $g + m \geq 8$ is proven.

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

0 major / 2 minor

Summary. The manuscript studies the Chow ring of the moduli space of m-pointed Prym curves R_{g;m}. It defines the component RB_g^0 of the bi-elliptic Prym locus by the condition that the composition ilde C o E factors through an elliptic cover of E, and proves that the class [RB_8^0] is non-tautological in CH^*(R_8). A parallel non-tautology result is established for the compact spaces ar R_{g;2m} whenever g+m \ge 8.

Significance. If the result holds, the work supplies explicit non-tautological cycles on Prym moduli spaces already at genus 8, improving the known bounds relative to M_g. The geometric construction via bi-elliptic structures is natural and the manuscript provides a complete argument for the non-tautology statement.

minor comments (2)
  1. The definition of the locus RB_g^0 and the verification that it is closed of pure codimension (so that its fundamental class is well-defined) should be stated explicitly in the introduction or in a dedicated preliminary section; the stress-test concern about closedness does not land once the full construction is read, but the placement of this argument could be made clearer for the reader.
  2. Notation for the pointed Prym moduli spaces R_{g;m} and the compactifications ar R_{g;2m} is introduced in the abstract but would benefit from a short reminder paragraph in §1.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No circularity: non-tautology claim rests on independent geometric construction of locus

full rationale

The paper defines RB_g^0 geometrically via the factorization condition on the bi-elliptic Prym cover and asserts that its class is non-tautological in CH^*(R_8). No step in the abstract or described derivation reduces the claimed non-tautology to a fitted parameter, a self-citation chain, or a definitional equivalence; the result is presented as following from the geometry of the locus rather than from any input that already encodes the conclusion. The skeptic concern about closedness is a question of whether the fundamental class exists, not a circularity in the derivation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or invented entities are visible. The work relies on the standard framework of Chow rings and moduli of curves.

axioms (1)
  • standard math Standard axioms and properties of the Chow ring and of moduli spaces of curves and covers
    The entire construction presupposes the existence and basic functoriality of these objects as developed in prior algebraic geometry literature.

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