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arxiv: 2605.21675 · v1 · 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 2026-05-22 08:18 UTC · model grok-4.3

classification 🧮 math.AG
keywords Prym moduli spacesnon-tautological cyclesChow ringbi-elliptic curvesetale double coverstautological subringmoduli of curves
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The pith

The fundamental class of the bi-elliptic Prym locus in genus 8 lies outside the tautological subring of the Chow ring of R_8.

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

This paper examines the Chow ring of the moduli space of Prym curves and shows that it contains classes beyond those generated by the usual tautological constructions. The central example is the locus RB_8^0 of bi-elliptic Prym curves, whose fundamental class is shown to be non-tautological. This gives non-tautological classes at genus 8, which is earlier than the known bounds for the ordinary moduli space of curves. The same approach yields non-tautological classes on certain compact pointed Prym moduli spaces when g plus the number of marked points is at least 8.

Core claim

The paper proves that the class [RB_8^0] is non-tautological in CH^*(R_8). The locus RB_g^0 parametrizes etale double covers of bi-elliptic curves such that the composition with the Prym cover factors through an elliptic cover of the base elliptic curve. This non-tautological property is obtained through explicit intersection computations that separate the class from the tautological subring, and a parallel result holds for the compact spaces overline{R}_{g;2m} whenever g + m is at least 8.

What carries the argument

The locus RB_g^0 of bi-elliptic Prym curves, the component where the bi-elliptic structure on the base curve composes with the etale double cover to factor through an elliptic cover of the elliptic curve.

If this is right

  • The Chow ring of R_8 properly contains its tautological subring.
  • Non-tautological classes exist in the Chow rings of the compact spaces overline{R}_{g;2m} for all g + m at least 8.
  • Prym moduli spaces admit non-tautological cycles at lower genus than the corresponding results for the moduli space of curves.
  • Explicit intersection theory on these loci can be used to produce further examples of non-tautological classes.

Where Pith is reading between the lines

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

  • Similar bi-elliptic or multi-elliptic constructions might produce non-tautological classes in Prym moduli spaces of genus less than 8.
  • The technique of factoring covers through elliptic curves could be adapted to other moduli spaces of covers to isolate additional cycles outside the tautological ring.
  • These classes may constrain the possible relations in the full Chow ring of Prym moduli spaces and suggest directions for computing its structure in low genus.

Load-bearing premise

The locus of bi-elliptic Prym curves RB_8^0 forms a well-defined closed subvariety whose fundamental class lies outside the tautological subring as detected by the intersection calculations.

What would settle it

An explicit intersection computation between [RB_8^0] and a chosen tautological class whose numerical value differs from the number predicted under the assumption that the class belongs to the tautological subring.

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 paper studies the Chow ring of the moduli space R_{g;m} of m-pointed Prym curves. It defines the locus RB_g^0 as the component of the bi-elliptic locus consisting of covers [C̃/C] such that the composition C̃ → E factors through an elliptic cover of the quotient elliptic curve E. The central claim is that the class [RB_8^0] is non-tautological in CH^*(R_8), proved by showing that this locus is a closed subvariety of expected codimension and that explicit intersection computations yield a numerical contradiction with any expression in the tautological generators. The same method produces non-tautological classes in the compactifications bar{R}_{g;2m} for all g + m ≥ 8.

Significance. If the result holds, it supplies explicit non-tautological cycles on Prym moduli spaces already at genus 8, improving the known bounds relative to the moduli space of curves M_g. The geometric construction via bi-elliptic structures and factorization through elliptic covers provides a concrete source of cycles outside the tautological subring and may extend to other Prym-related moduli problems.

minor comments (2)
  1. A brief table or list summarizing the pairs (g, m) for which non-tautological classes are obtained in bar{R}_{g;2m} would improve readability of the main theorem.
  2. The notation for the various compactifications (R_g, bar{R}_{g;2m}, etc.) is consistent but could be collected in a single preliminary subsection for quick reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive report, accurate summary of our results on the non-tautological class of the bi-elliptic Prym locus in the Chow ring of the Prym moduli space, and recommendation for minor revision. We appreciate the recognition that our geometric construction via bi-elliptic structures provides explicit non-tautological cycles at genus 8, improving upon known bounds for M_g. We will incorporate minor revisions to clarify any points and improve the exposition in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified in the derivation

full rationale

The paper defines the locus RB_g^0 geometrically as the component of bi-elliptic Prym curves where the composition factors through an elliptic cover, then uses explicit intersection computations on the Prym moduli space to derive a numerical contradiction with any hypothetical tautological expression. These steps rely on standard properties of moduli spaces of curves and Prym varieties that are independent of the target non-tautological claim, with no reduction to self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations. The argument is self-contained and externally verifiable via geometric constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard facts from intersection theory on moduli spaces and the definition of the bi-elliptic locus; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • standard math Standard properties of Chow rings and tautological subrings on moduli spaces of curves and covers hold.
    Invoked implicitly when distinguishing tautological from non-tautological classes.

pith-pipeline@v0.9.0 · 5806 in / 1220 out tokens · 32553 ms · 2026-05-22T08:18:39.191758+00:00 · methodology

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