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Prym varieties and projective structures on Riemann surfaces

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that the canonical projective structure on a Riemann surface built from a Prym variety genuinely depends on the chosen étale double cover, so it is a new object on the moduli space of covers rather than on moduli of curves.

desk verdict New Prym-theoretic projective structures with a clean derivative computation; the non-descent proof has a repairable but real gap in Lemma 5.8. read the letter →

arxiv 2506.02871 v1 pith:I6TXVLJ4 submitted 2025-06-03 math.AG

Indranil Biswas , Alessandro Ghigi , Luca Vai This is my paper
classification math.AG MSC 14H1014H4014K2553B10
keywords PrymvarietyprojectivestructureétaledoublecoverthetafunctionsSchottky–JungidentitymodulispaceofcurvesThetanullwertmap∂-derivative
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An étale double cover of a Riemann surface carries a Prym variety, and the paper uses that Prym variety to construct two canonical projective structures—coordinate atlases with Möbius transition functions—one on the cover and one on the base curve. The main question is whether the projective structure on the base curve depends on the chosen cover. The paper computes the ∂-derivative of this structure as a pullback of the Fubini–Study metric through the Prym period map and the Thetanullwert map, then uses the Schottky–Jung identities to show that this derivative does not descend from the moduli space of covers to the moduli space of curves. The result is a negative answer: there are curves with two distinct double covers giving different Prym projective structures. This matters because it produces a genuinely new canonical object that lives on the moduli space of covers, unlike the uniformization, Hodge, and theta structures previously known.

What carries the argument

The central object is the Prym variety P = (ker Nm_π)^0 of the double cover, with its principal polarization Ξ, together with the intrinsically defined line L ⊂ $H^{0}$(P, 2Ξ) spanned by the unique section orthogonal to the sections vanishing at the origin. The Prym difference map φ(p,q) = O(p − σ(p) − q + σ(q)) pulls 2Ξ back to K_{C~×C~}(2Δ~ − 2Σ), where Σ is the graph of the deck involution σ; normalizing that pullback along the diagonal gives the projective structure. The derivative computation runs through Lemma 4.2, which expresses the restriction of the pullback to 3Δ in terms of the value and second derivatives at the origin, then through the heat equation for second-order $\theta$ functions, and finally through the Schottky–Jung identities that relate the Prym period matrix to the period matrix of the base curve.

What would settle it

Compute the full Sp(2g,Z) stabilizer of e_{y0} = diag(y0, Π') and check whether its induced action on the tangent space T_{e_{y0}}(H_1 × {Π'}) is trivial; if some stabilizer element moves the tangent generator w, the non-descent proof collapses at that point, while if all act trivially Lemma 5.8's conclusion is saved despite the inaccurate automorphism statement.

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Extended reading notes

Core claim

The paper establishes that the Prym projective structure β^P_π on the base curve C genuinely remembers the étale double cover π : C~ → C. Theorem 1.3 computes ∂β^P = 8π P_g^* Θ_{g-1}^* ω_{FS} on the moduli space R_g of étale double covers, where P_g is the Prym map and Θ_{g-1} is the second-order Thetanullwert map. Through the Schottky–Jung identities this becomes 8π Π^*(Θ'_g)^*ω_{FS} on Torelli space, where Π is the period map and Θ'_g is another Thetanullwert map. The form (Θ'_g)^*ω_{FS} restricts to zero on one product slice H_1 × {Π'} of the Siegel space and to an immersion on the swapped slice {Π'} × H_1; since a symplectic transformation swaps the two slices, the form cannot be Sp(2g,Z)-invariant and does not descend to M_g. Consequently β^P is not pulled back from M_g, and Corollary 5.10 states that two distinct étale covers of the same curve can yield different Prym projective structures.

Load-bearing premise

The negative answer to Question 1.1 rests on Lemma 5.8, which assumes that a product period point E_{y0} × J(C') with generic factors has trivial symplectic stabilizer; the product actually has at least the sign automorphisms, so the proof as written needs the replacement fact—that every stabilizer element acts trivially on the slice H_1 × {Π'}—which the paper does not establish.

Editorial extensions

If this is right

  • The Prym projective structure β^P is a new kind of canonical object: a section of the affine bundle over the moduli space of étale double covers that is not pulled back from the moduli space of curves.
  • Since the uniformization, Hodge, and theta projective structures do descend to M_g, β^P is genuinely different from all of them.
  • The ∂-derivative computation gives a practical criterion for detecting whether a family of projective structures descends to moduli, namely whether its Thetanullwert pullback form is Sp(2g,Z)-invariant.
  • Question 1.2, about whether the projective structure on the cover C~ depends on the involution, remains open and is not settled by the same descent argument.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same ∂-derivative technique could potentially answer Question 1.2 if a suitable degeneration of the covers themselves is found, though the paper states that this is not clear.
  • The non-descent of the Thetanullwert form suggests that Θ'_g carries nontrivial moduli-theoretic information on Torelli space that can obstruct pullbacks of sections; this may extend to other Prym-type or Prym–Tyurin constructions.
  • One could test Corollary 5.10 explicitly in low genus by computing the two projective structures on a specific curve with two double covers and comparing their Schwarzian derivatives; the paper's proof establishes existence via non-descent rather than an explicit pair.
  • The imprecision in Lemma 5.8 about automorphisms of a product variety may be repairable: the conclusion only needs the stabilizer elements to act trivially on the slice H_1 × {Π'}, not the full automorphism group to be trivial.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. Given an étale double cover π: C~→C of compact Riemann surfaces with C of genus at least two, the authors construct projective structures β^Q_π on C~ and β^P_π on C from the Prym variety P(π) and the canonically defined line L⊂H^0(P,2Ξ). They show that β^Q_π is σ-invariant and descends to C, and they interpret β^P_π as a function of the cover. Working over the Torelli space, they view β^P as a section of an affine bundle and compute its ∂-derivative as 8π τ^*Θ_{g-1}^*ω_FS, where τ is the Prym period map and Θ_{g-1} the Thetanullwert map. Using the Schottky–Jung identities they rewrite this as 8π Π^*(Θ'_g)^*ω_FS, with Π the period map of the base curve. They then prove by a degeneration argument that this (1,1)-form does not descend to M_g, and conclude that β^P depends on the covering, giving a negative answer to Question 1.1. Question 1.2 is left open.

Significance. The main computation in Section 4 is detailed and essentially parameter-free: the line L is intrinsic, the heat equation determines ∂β^P, and no fitted input is used. The passage through the Schottky–Jung identities in Section 5 is elegant and yields a concrete formula for ∂β^P on the Torelli space. If the degeneration lemma is repaired, the negative answer to Question 1.1 is a substantial result, since it exhibits a canonical projective structure whose moduli behavior genuinely distinguishes the covering data, in contrast with the uniformization, Hodge, and theta structures. The paper also gives a useful explicit codifferential of the Prym period map in Proposition 3.2. However, the proof of the descent statement rests on a boundary argument that currently contains a false automorphism-group assertion, so the main claim is not yet fully established.

major comments (2)
  1. [5.2, Lemma 5.8] The assertion that 'By the assumptions on y0 and J(C') the abelian variety Ey0 × J(C') does not have automorphisms' is false. A product of two positive-dimensional abelian varieties always contains at least the automorphism (−1,−1), and the period point ey0 = diag(y0,Π') is fixed by −I_{2g} in Sp(2g,Z); more generally the stabilizer contains block-diagonal elements diag(±I2,±I_{2g−2}). Consequently the proof of (5.22) is incomplete: the Baire argument yields only γ ∈ Stab(ey0), and the step 'Hence γ = 1' is not justified. Since (5.22) is exactly what identifies the boundary tangent vector w with a limit of period-map images of tangent vectors from the smooth Torelli space, this is load-bearing for Proposition 5.9 and Corollary 5.10. The likely repair is to prove that, for generic y0 and for the chosen simple non-isogenous J(C'), every element of Stab(ey0) acts trivially on the slice H1×{Π'}; this is plausible but is not what the paper states and needs an explicit argument.
  2. [5.2, Proposition 5.9] The passage from 'the last equation is true for generic w ∈ T(H1×{Π'})' to the displayed equality of restrictions is notationally compressed. What the limit argument establishes is an equality of (1,1)-forms on H1×{Π'} at the point ey0, namely (Θ')^*ω_FS|_{H1×{Π'}} = M^*( (Θ')^*ω_FS|_{{Π'}×H1}) after identifying the two slices via M. The displayed line, which appears to equate a form on {Π'}×H1 with a form on H1×{Π'}, should be rewritten to make the role of the isomorphism induced by M explicit. The intended contradiction is clear, but the current wording is ambiguous in a load-bearing paragraph and should be corrected in the revision.
minor comments (5)
  1. [2.3, Proposition 2.10] In the statement, 'H^0(eS, KeS(2eS −2Σ))' should read 'H^0(eS, KeS(2e∆ −2Σ))'.
  2. [2.3, Corollary 2.11] The phrase 'By 2.10' should be 'By Proposition 2.10'.
  3. [5.2, Lemma 5.8] The expression '$B \subset M g − \Delta_0$' should be '$B \subset M_g \setminus \Delta_0$'.
  4. [4.1, Lemma 4.2] In the proof, the phrase 'by the mean value theorem' is not accurate; the argument uses a first-order Taylor expansion of the map φ∘e.
  5. [5.1, Corollary 5.4] The same symbol ω_FS is used for Fubini–Study metrics on different projective spaces; although the authors announce this convention, indicating the target projective space in each pullback would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivative computation is parameter-free and the central non-descent argument rests on the classical Schottky–Jung identity and independent degeneration analysis, not on the paper's own conclusions.

full rationale

The derivation chain is not circular. The Prym projective structure is canonically defined from the intrinsically defined line L in H^0(P,2Ξ), and the derivative formula in Theorem 4.3 is obtained by a direct local computation (Lemma 4.2) plus the heat equation for theta functions and the Riemann summation formula; no fitted parameter is introduced and no 'prediction' is fed back as an input. The passage to the base curve uses the Schottky–Jung identity from [FR], an external classical theorem, and the comparison of Thetanullwert maps in Proposition 5.2 is proved in the paper using the Riemann summation formula, with only an elementary unitarity check delegated to [BV]. The negative answer to Question 1.1 is inferred from non-descent of the computed (1,1)-form to M_g, a logically valid contrapositive: if βP descended, its ∂-derivative would descend. The non-descent argument does depend on Lemma 5.8, and the paper's assertion there that E_y0 × J(C') 'does not have automorphisms' is inaccurate because such a product always has at least the automorphism (−1,−1). This is a genuine correctness gap or repairable glitch in the proof, but it is not circular: the conclusion of Lemma 5.8 is not assumed as input, and even the needed repair would use an independent stabilizer computation rather than a result equivalent to the target statement. The self-citations to [BGV] and [BV] are to parameter-free, published or preprint mathematical arguments with independent proofs, and they are not used as unverified premises that determine the answer. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The construction introduces no free parameters and no new objects beyond the standard Prym variety, theta groups, and Thetanullwert maps. The main burden rests on classical results (Prym theory, theta identities, Schottky-Jung) and on one technical degeneration lemma whose stated automorphism claim appears imprecise.

assumptions (5)
  • standard math Prym variety associated to an etale double cover carries a principal polarization Ξ.
    Used throughout Section 2.2 to define the line L and the construction.
  • standard math The theta functions {θ_u} form an orthogonal basis of V(τ) with constant length, and satisfy the heat equation.
    Invoked in Theorem 4.3 via [vG] and [Ig].
  • standard math Schottky-Jung identities relate the Prym period matrix to the base period matrix.
    Used in Theorem 5.1 and Corollary 5.4, cited from [FR].
  • standard math The Fubini-Study metric and the Veronese embedding satisfy v2*ω = 2ω.
    Used in Proposition 5.2 and Corollary 5.4, proven in [BV, Lemma 3.1].
  • ad hoc to paper The automorphism group statement in Lemma 5.8: E_y0 × J(C') has no automorphisms and its Sp-stabilizer is trivial.
    This is asserted but appears inaccurate for a product of generic factors; the proof may be repairable if all stabilizer elements act trivially on the slice H1×{Π'}.

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Pith. "Pith review of Prym varieties and projective structures on Riemann surfaces." pith.science (2026). https://pith.science/paper/I6TXVLJ4

@misc{pith2026250602871,
  author       = {Pith},
  title        = {Pith review of: Prym varieties and projective structures on Riemann surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I6TXVLJ4}},
  note         = {Machine review of arXiv:2506.02871}
}
abstract

Given an \'etale double covering $\pi\, :\, \widetilde{C}\, \longrightarrow\, C$ of compact Riemannsurfaces with $C$ of genus at least two, we use the Prym variety of the cover to construct canonical projective structures on both $\widetilde C$ and $C$. This construction can be interpreted as a section of an affine bundle over the moduli space of \'etale double covers. The $\overline{\partial}$--derivative of this section is a (1,1)--form on the moduli space. We compute this derivative in terms of Thetanullwert maps. Using the Schottky--Jung identities we show that, in general, the projective structure on $C$ depends on the cover.

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Works this paper leans on

29 extracted references · 28 canonical work pages

  1. [1]

    Arbarello, M

    E. Arbarello, M. Cornalba and P. B. Griffiths, Geometry of algebraic curves: volume II with a contribution by Joseph Daniel Harris, Vol. 268. Springer Science and Business Media, 2011

    work page 2011

  2. [2]

    Beauville, Prym varieties: a survey

    A. Beauville, Prym varieties: a survey. Proc. of Symp. in Pure Mat. Vol. 49. No. 1. 1989

    work page 1989

  3. [3]

    Beauville, Variétés de Prym et jacobiennes intermédiaires , Annales scientifiques de l'École Normale Supérieure

    A. Beauville, Variétés de Prym et jacobiennes intermédiaires , Annales scientifiques de l'École Normale Supérieure. Vol. 10. No. 3. 1977

    work page 1977

  4. [4]

    Beauville and O

    A. Beauville and O. Debarre. Sur le probleme de Schottky pour les variétés de Prym , Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 14.4 (1987): 613-623

    work page 1987

  5. [5]

    Biswas, A

    I. Biswas, A. Ghigi and C. Tamborini, Theta bundle, Quillen connection and the Hodge theoretic projective structure , Commun. Math. Phys. (2024) 405:243

    work page 2024

  6. [6]

    Birkenhake and H

    C. Birkenhake and H. Lange, Complex abelian varieties , volume 302 of Grundlehren der Mathematischen Wissenschaften . Springer-Verlag, Berlin, second edition, 2004

    work page 2004

  7. [7]

    Biswas, E

    I. Biswas, E. Colombo, P. Frediani and G. P. Pirola, Hodge theoretic projective structure on Riemann surfaces, Jour. Math. Pures. Appl. 149 (2021), 1--27

    work page 2021

  8. [8]

    Biswas, F

    I. Biswas, F. F. Favale, G. P. Pirola and S. Torelli, Quillen connection and the uniformization of Riamann surfaces , Annali di Matematica Pura ed Applicata 201 (2022), 2825--2835

    work page 2022

Show all 29 references
  1. [9]

    Biswas, A

    I. Biswas, A. Ghigi and L. Vai, Theta functions and projective structures , Int. Math. Res. Not. Vol. 2025, Issue 1, rnae271

    2025

  2. [10]

    Biswas and L

    I. Biswas and L. Vai, On the projective structures given by theta, Arxiv [Preprint] (2025). Arxiv, https://arxiv.org/abs/2504.15866

    2025 arXiv

  3. [11]

    Biswas and A

    I. Biswas and A. K. Raina, Projective structures on a Riemann surface, Inter. Math. Res. Not. (1996), No. 15, 753--768

    1996

  4. [12]

    J. D. Fay, Theta functions on Riemann surfaces, Vol. 352. Springer, 2006

    2006

  5. [13]

    H. M. Farkas and H. E. Rauch, Period relations of Schottky type on Riemann surfaces, Ann. of Math. 92 (1970), 434--461

    1970

  6. [14]

    Igusa, Theta functions , Vol

    J.-I. Igusa, Theta functions , Vol. 194. Springer Science and Business Media, 2012

    2012

  7. [15]

    P. B. Griffiths and J. Harris, Principles of algebraic geometry , John Wiley and Sons, 2014

    2014

  8. [16]

    Grushevsky and R

    S. Grushevsky and R. Salvati Manni, Gradients of odd theta functions, Journal reine ang. Math. 573 (2004), 45--59

    2004

  9. [17]

    R. C. Gunning, Special coordinate coverings of Riemann surfaces, Math. Ann. 170 (1967), 67--86

    1967

  10. [18]

    R. C. Gunning, Lectures on Riemann surfaces , Princeton University Press, 1976

    1976

  11. [19]

    J. H. Hubbard, The monodromy of projective structures, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), 257--275, Ann. of Math. Stud., Vol. 97, Princeton Univ. Press, Princeton, N.J. 1981

    1978

  12. [20]

    Izadi, Second order theta divisors on Pryms, Bull

    E. Izadi, Second order theta divisors on Pryms, Bull. Soc. Math. Fr. 127 (1999), 1--23

    1999

  13. [21]

    Izadi and C

    E. Izadi and C. Pauly, Some Properties of Second Order Theta Functions o Prym Varieties, Math. Nach. 230 (2001), 73--91

    2001

  14. [22]

    Mumford, Prym varieties I, Contributions to analysis , pp

    D. Mumford, Prym varieties I, Contributions to analysis , pp. 325--350, Academic Press, 1974

    1974

  15. [23]

    Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol

    D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, American Mathematical Society, 2008

    2008

  16. [24]

    J. C. Naranjo, Fourier transform and Prym varieties, Jour. Reine Ang. Math. (2003), 221-230

    2003

  17. [25]

    Poor, Fay's trisecant formula and cross-ratios, Proc

    C. Poor, Fay's trisecant formula and cross-ratios, Proc. Amer Math Soc 114 (1992), 667--671

    1992

  18. [26]

    Salvati Manni, Modular varieties with level 2 theta structure, Amer

    R. Salvati Manni, Modular varieties with level 2 theta structure, Amer. Jour. Math. 116 (1994), 1489--1511

    1994

  19. [27]

    A. N. Tyurin, On periods of quadratic differentials, Russian Math. Surveys 33 (1987), 169--221

    1987

  20. [28]

    van Geemen, The Schottky problem and second order theta functions, Workshop on Abelian Varieties and Theta Functions (Spanish) (Morelia, 1996), 1998

    B. van Geemen, The Schottky problem and second order theta functions, Workshop on Abelian Varieties and Theta Functions (Spanish) (Morelia, 1996), 1998

    1996

  21. [29]

    P. G. Zograf and L. A. Takhtadzhyan, On the uniformization of Riemann surfaces and on the Weil-Petersson metric on the Teichm\"uller and Schottky spaces, Math. USSR-Sb. 60 (1988), 297--313

    1988

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