pith. sign in

arxiv: 2207.08966 · v5 · pith:B4ZLFN53new · submitted 2022-07-18 · 🧮 math.AC · math.AG

Finite F-representation type for homogeneous coordinate rings of non-Fano varieties

Devlin Mallory This is my paper

Pith reviewed 2026-05-24 11:29 UTC · model grok-4.3

classification 🧮 math.AC math.AG
keywords finite F-representation typehomogeneous coordinate ringspositive characteristicdifferential operatorscotangent sheafabelian varietiesCalabi-Yau varietiescomplete intersections
0
0 comments X

The pith

Homogeneous coordinate rings of abelian varieties, most Calabi-Yau varieties, and complete intersections of general type fail to have finite F-representation type in positive characteristic.

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

The paper aims to show that many homogeneous coordinate rings of varieties in positive characteristic do not have finite F-representation type. It establishes a connection between the differential operators on these rings and global sections of twists of the dual of symmetric powers of the cotangent sheaf. When that sheaf is not positive, this allows ruling out the finite property. This applies to classes like abelian varieties, most Calabi-Yau varieties, and complete intersections of general type, providing many explicit examples where the property fails.

Core claim

A connection is proved between differential operators on the homogeneous coordinate ring of a variety X and the existence of global sections of a twist of the dual of symmetric powers of its cotangent sheaf. This connection, combined with positivity conditions, shows that the coordinate rings of abelian varieties, most Calabi-Yau varieties, and complete intersections of general type fail to have finite F-representation type.

What carries the argument

The correspondence between differential operators on the homogeneous coordinate ring and global sections of twists of the dual of symmetric powers of the cotangent sheaf.

Load-bearing premise

That non-positivity of the twisted dual of symmetric powers of the cotangent sheaf implies the non-existence of finite F-representation type through the established connection to differential operators.

What would settle it

A direct calculation showing that the homogeneous coordinate ring of an abelian variety in positive characteristic does have finite F-representation type.

read the original abstract

Finite $F$-representation type is an important notion in characteristic-$p$ commutative algebra, but explicit examples of varieties with or without this property are few. We prove that a large class of homogeneous coordinate rings in positive characteristic will fail to have finite $F$-representation type. To do so, we prove a connection between differential operators on the homogeneous coordinate ring of $X$ and the existence of global sections of a twist of $(\mathrm{Sym}^m \Omega_X)^\vee$. By results of Takagi and Takahashi, this allows us to rule out FFRT for coordinate rings of varieties with $(\mathrm{Sym}^m \Omega_X)^\vee$ not ``positive''. By using results positivity and semistability conditions for the (co)tangent sheaves, we show that several classes of varieties fail to have finite $F$-representation type, including abelian varieties, most Calabi--Yau varieties, and complete intersections of general type. Our work also provides examples of the structure of the ring of differential operators for non-$F$-pure varieties, which to this point have largely been unexplored.

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

1 major / 1 minor

Summary. The paper claims to prove that homogeneous coordinate rings of a large class of non-Fano varieties over fields of positive characteristic—including abelian varieties, most Calabi-Yau varieties, and complete intersections of general type—fail to have finite F-representation type. The argument proceeds by establishing a correspondence between the ring of differential operators on the homogeneous coordinate ring R and the existence of global sections of positive twists of (Sym^m Ω_X)^∨; when the latter sheaf is not positive, results of Takagi and Takahashi are invoked to conclude the absence of FFRT. The work also supplies information on the structure of D(R) in the non-F-pure setting.

Significance. If the central correspondence and its applicability hold, the result would substantially enlarge the known supply of explicit examples without finite F-representation type and would furnish the first systematic information on D(R) for non-F-pure rings, a case previously largely unexplored. The geometric criterion based on positivity and semistability of cotangent sheaves offers a potentially reusable bridge between algebraic and geometric invariants in characteristic p.

major comments (1)
  1. [the connection between differential operators and global sections of twists of (Sym^m Ω_X)^∨ together with the statement] The invocation of Takagi-Takahashi theorems to rule out FFRT when (Sym^m Ω_X)^∨ is not positive is stated to apply to the listed classes, yet those theorems are formulated for F-pure rings. The manuscript explicitly includes non-F-pure examples (abelian varieties, most Calabi-Yau varieties) among those claimed to lack FFRT; it is therefore necessary to verify whether the correspondence or the Takagi-Takahashi step requires an F-purity hypothesis that is absent from these classes. This point is load-bearing for the central claim.
minor comments (1)
  1. Notation for the dual sheaf (Sym^m Ω_X)^∨ and the precise range of m for which the correspondence is proved should be stated uniformly in the introduction and in the main theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying this key point about the hypotheses of the Takagi-Takahashi theorems. We address the concern below and will revise the manuscript to clarify the logic.

read point-by-point responses

  1. Referee: The invocation of Takagi-Takahashi theorems to rule out FFRT when (Sym^m Ω_X)^∨ is not positive is stated to apply to the listed classes, yet those theorems are formulated for F-pure rings. The manuscript explicitly includes non-F-pure examples (abelian varieties, most Calabi-Yau varieties) among those claimed to lack FFRT; it is therefore necessary to verify whether the correspondence or the Takagi-Takahashi step requires an F-purity hypothesis that is absent from these classes. This point is load-bearing for the central claim.

    Authors: The referee is correct that the theorems of Takagi and Takahashi require F-purity. The correspondence we establish between differential operators on R and global sections of twists of (Sym^m Ω_X)^∨ holds without any F-purity hypothesis. For the F-pure varieties among the classes considered, the application of Takagi-Takahashi proceeds as stated. For the non-F-pure examples (such as abelian varieties and most Calabi-Yau varieties), the manuscript instead uses the correspondence to give an explicit description of the structure of D(R) and invokes this description to conclude the absence of FFRT, independently of Takagi-Takahashi. We will revise the manuscript to separate the F-pure and non-F-pure cases explicitly and to detail the argument used for the non-F-pure setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external independent theorems

full rationale

The paper derives a connection between the ring of differential operators on the homogeneous coordinate ring R and global sections of twists of (Sym^m Ω_X)^∨, then applies external results of Takagi-Takahashi (distinct authors) to rule out FFRT when the sheaf lacks positivity, combined with standard positivity/semistability theorems from the literature. No step reduces the central claim to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The cited theorems are treated as independent external benchmarks, and the argument does not import uniqueness or ansatzes from the authors' prior work. This is the normal case of a paper building on prior literature without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on domain assumptions from algebraic geometry and two cited prior theorems; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract description.

axioms (2)
  • domain assumption Results of Takagi and Takahashi connecting differential operators to finite F-representation type via global sections
    Invoked in the abstract to translate non-positivity of the sheaf into failure of finite F-representation type.
  • domain assumption Positivity and semistability conditions hold for the cotangent sheaves of abelian varieties, most Calabi-Yau varieties, and general type complete intersections
    Used to conclude that the relevant twisted dual symmetric powers have no global sections.

pith-pipeline@v0.9.0 · 5719 in / 1514 out tokens · 35732 ms · 2026-05-24T11:29:47.531048+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Achinger, A characterization of toric varieties in characteristic p , Int.\ Math.\ Res.\ Not.\ IMRN (2015), no

    P. Achinger, A characterization of toric varieties in characteristic p , Int.\ Math.\ Res.\ Not.\ IMRN (2015), no. 16, 6879--6892

    work page 2015

  2. [2]

    Artin, Supersingular K3 surfaces , Ann.\ Sci.\ \' E cole Norm.\ Sup

    M. Artin, Supersingular K3 surfaces , Ann.\ Sci.\ \' E cole Norm.\ Sup. (4) 7 (1974), 543--567

    work page 1974

  3. [3]

    Bogomolov and B

    F. Bogomolov and B. De Oliveira, Symmetric tensors and geometry of P^N subvarieties , Geom.\ Funct.\ Anal.\ 18 (2008), no. 3, 637--656

    work page 2008

  4. [4]

    Buchweitz, D

    R.-O. Buchweitz, D. Eisenbud, and J. Herzog, Cohen- M acaulay modules on quadrics , in: Singularities, representation of algebras, and vector bundles ( L ambrecht, 1985), pp. 58--116, Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987

    work page 1985

  5. [5]

    Dao and P.\,H

    H. Dao and P.\,H. Quy, On the associated primes of local cohomology, Nagoya Math. J.\ 237 (2020), 1--9

    work page 2020

  6. [6]

    Deligne and N

    P. Deligne and N. Katz (eds), S\'eminaire de G\'eom\'etrie alg\'ebrique du Bois-Marie 1967--1969 (SGA 7 II), Groupes de monodromie en g\'eom\'etrie alg\'ebrique. II, Lecture Notes in Math., vol. 340, Springer-Verlag, Berlin-New York , 1973

    work page 1967

  7. [7]

    Dufresne, Separating invariants and finite reflection groups, Adv.\ Math.\ 221 (2009), no

    E. Dufresne, Separating invariants and finite reflection groups, Adv.\ Math.\ 221 (2009), no. 6, 1979--1989

    work page 2009

  8. [8]

    Hara, Looking out for frobenius summands on a blown-up surface of P2 , Illinois J

    N. Hara, Looking out for frobenius summands on a blown-up surface of P2 , Illinois J. Math.\ 59 (2015), 115--142

    work page 2015

  9. [9]

    Hara and R

    N. Hara and R. Ohkawa, The FFRT property of two-dimensional normal graded rings and orbifold curves , Adv.\ Math.\ 370 (2020), 107215

    work page 2020

  10. [10]

    Hartshorne, Algebraic Geometry , Grad.\ Texts in Math., vol

    R. Hartshorne, Algebraic Geometry , Grad.\ Texts in Math., vol. 52, Springer, New York, NY, 1977

    work page 1977

  11. [11]

    Hochster and L

    M. Hochster and L. N\' u \ n ez -Betancourt, Support of local cohomology modules over hypersurfaces and rings with FFRT , Math.\ Res.\ Lett.\ 24 (2017), no. 2, 401--420

    work page 2017

  12. [12]

    Hsiao, A remark on bigness of the tangent bundle of a smooth projective variety and D -simplicity of its section rings , J.\ Algebra Appl.\ 14 (2015), no

    J.-C. Hsiao, A remark on bigness of the tangent bundle of a smooth projective variety and D -simplicity of its section rings , J.\ Algebra Appl.\ 14 (2015), no. 7, 1550098

    work page 2015

  13. [13]

    Jeffries, Derived functors of differential operators, Int.\ Math.\ Res.\ Not.\ IMRN (2021), no

    J. Jeffries, Derived functors of differential operators, Int.\ Math.\ Res.\ Not.\ IMRN (2021), no. 7, 4920--4940

    work page 2021

  14. [14]

    Ji, The N oether-- L efschetz theorem , preprint arXiv:2107.12962 (2021)

    L. Ji, The N oether-- L efschetz theorem , preprint arXiv:2107.12962 (2021). To appear in J. Algebraic Geom

    work page arXiv 2021

  15. [15]

    Lange and C

    H. Lange and C. Pauly, On F robenius-destabilized rank-2 vector bundles over curves , Comment.\ Math.\ Helv.\ 83 (2008), no. 1, 179--209

    work page 2008

  16. [16]

    Langer, Moduli spaces of sheaves and principal G -bundles , in: Algebraic geometry--- S eattle 2005

    A. Langer, Moduli spaces of sheaves and principal G -bundles , in: Algebraic geometry--- S eattle 2005. P art 1 , pp. 273--308, Proc.\ Sympos.\ Pure Math., vol. 80, Amer.\ Math.\ Soc., Providence, RI, 2009

    work page 2005

  17. [17]

    , Generic positivity and foliations in positive characteristic, Adv.\ Math.\ 277 (2015), 1--23

    work page 2015

  18. [18]

    Mallory, Bigness of the tangent bundle of del P ezzo surfaces and D -simplicity , Algebra Number Theory 15 (2021), no

    D. Mallory, Bigness of the tangent bundle of del P ezzo surfaces and D -simplicity , Algebra Number Theory 15 (2021), no. 8, 2019--2036

    work page 2021

  19. [19]

    To appear in Ill

    , Homogeneous coordinate rings as direct summands of regular rings, preprint arXiv:2206.03621 (2022). To appear in Ill. J. Math

    work page arXiv 2022

  20. [20]

    Noma, Stability of F robenius pull-backs of tangent bundles and generic injectivity of G auss maps in positive characteristic , Compos.\ Math.\ 106 (1997), no

    A. Noma, Stability of F robenius pull-backs of tangent bundles and generic injectivity of G auss maps in positive characteristic , Compos.\ Math.\ 106 (1997), no. 1, 61--70

    work page 1997

  21. [21]

    , Stability of F robenius pull-backs of tangent bundles of weighted complete intersections , Math.\ Nachr.\ 221 (2001), 87--93

    work page 2001

  22. [22]

    Raedschelders, S

    T. Raedschelders, S . S penko, and M. Van den Bergh, The F robenius morphism in invariant theory II , Adv. Math. 410 Part A (2022), article ID 108587

    work page 2022

  23. [23]

    Ramanan and A

    S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math.\ J.\ (2) 36 (1984), no. 2, 269--291

    work page 1984

  24. [24]

    Sannai and H

    A. Sannai and H. Tanaka, A characterization of ordinary abelian varieties by the F robenius push-forward of the structure sheaf , Math.\ Ann.\ 366 (2016), no. 3-4, 1067--1087

    work page 2016

  25. [25]

    Shibuta, One-dimensional rings of finite F -representation type , J

    T. Shibuta, One-dimensional rings of finite F -representation type , J. Algebra 332 (2011), 434--441

    work page 2011

  26. [26]

    Shioda, An example of unirational surfaces in characteristic p , Math.\ Ann.\ 211 (1974), 233--236

    T. Shioda, An example of unirational surfaces in characteristic p , Math.\ Ann.\ 211 (1974), 233--236

    work page 1974

  27. [27]

    2, 153--168

    , Some results on unirationality of algebraic surfaces, Math.\ Ann.\ 230 (1977), no. 2, 153--168

    work page 1977

  28. [28]

    Singh and I

    A.\,K. Singh and I. Swanson, Associated primes of local cohomology modules and of F robenius powers , Int.\ Math.\ Res.\ Not.\ (2004), no. 33, 1703--1733

    work page 2004

  29. [29]

    Smith, The D -module Structure of F - Split Rings , Math.\ Res.\ Lett

    K.\,E. Smith, The D -module Structure of F - Split Rings , Math.\ Res.\ Lett. 2 (1995), no. 4, 377--386

    work page 1995

  30. [30]

    Smith and M

    K.\,E. Smith and M. Van den Bergh, Simplicity of rings of differential operators in prime characteristic, Proc.\ London Math.\ Soc. (3) 75 (1997), no. 1, 32--62

    work page 1997

  31. [31]

    Takagi and R

    S. Takagi and R. Takahashi, D -modules over rings with finite F -representation type , Math.\ Res.\ Lett.\ 15 (2008), no. 3, 563--581

    work page 2008

  32. [32]

    Tango, On the behavior of extensions of vector bundles under the F robenius map , Nagoya Math

    H. Tango, On the behavior of extensions of vector bundles under the F robenius map , Nagoya Math. J.\ 48 (1972), 73--89

    work page 1972

  33. [33]

    Thomsen, Frobenius direct images of line bundles on toric varieties, J

    J.\,F. Thomsen, Frobenius direct images of line bundles on toric varieties, J. Algebra 226 (2000), no. 2, 865--874

    work page 2000

  34. [34]

    Yekutieli, An explicit construction of the G rothendieck residue complex (with an appendix by P

    A. Yekutieli, An explicit construction of the G rothendieck residue complex (with an appendix by P. Sastry), Ast\' e risque 208 (1992)

    work page 1992