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arxiv: 2605.16636 · v1 · pith:2F6V2QYNnew · submitted 2026-05-15 · 🧮 math.AG · math.AC

On positivity of the limit F-signature

Yuchen Liu , Suchitra Pande This is my paper

Pith reviewed 2026-05-19 20:59 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords F-signatureKLT singularitiespositive characteristicbirational geometrytoric degenerationsK-stabilityFrobenius-alpha invariant
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The pith

The limit of F-signatures for KLT singularities stays bounded away from zero in three dimensions for non-weakly exceptional cases and for smooth hypersurfaces of very low degree.

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

The paper establishes that a conjecture of Carvajal-Rojas, Schwede and Tucker holds in two families of cases: the F-signatures of reductions of complex KLT singularities to large positive characteristic p remain bounded away from zero. This is shown first for three-dimensional non-weakly exceptional singularities via an inductive argument that reduces the problem using birational models. The same positivity is shown for smooth hypersurfaces of very low degree by building isotrivial normal toric degenerations that preserve the relevant invariants. A sympathetic reader cares because the result gives concrete evidence that singularities behave well when passing from characteristic zero to positive characteristic, and the methods draw on ideas from K-stability to control the limit uniformly in p.

Core claim

We prove that this conjecture holds for three-dimensional non-weakly exceptional singularities by an inductive argument. We also prove that the conjecture holds for smooth hypersurfaces of very low degree by constructing isotrivial normal toric degenerations. By considering the version of this conjecture for the Frobenius-alpha invariant, our techniques are inspired by K-stability theory and involve using degenerations and birational geometry.

What carries the argument

Inductive reduction via birational models together with isotrivial normal toric degenerations, used to keep F-signature behavior uniform across all large primes p.

If this is right

  • The conjecture on positive limit F-signature holds for all three-dimensional non-weakly exceptional KLT singularities.
  • Smooth hypersurfaces of very low degree satisfy the same positivity statement after reduction to large characteristic.
  • The same methods apply directly to the Frobenius-alpha invariant version of the conjecture.
  • K-stability techniques can be used to produce degenerations that preserve lower bounds on F-signature.

Where Pith is reading between the lines

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

  • The inductive and degeneration methods may suggest a route toward the full conjecture if analogous models exist in higher dimensions.
  • These results connect the positivity question to existing birational geometry tools that already control other invariants under reduction modulo p.
  • The toric degeneration construction could be tested on other classes of hypersurfaces to see whether the very-low-degree restriction can be relaxed.

Load-bearing premise

The arguments require the existence of suitable birational models or degenerations in which the F-signature can be controlled uniformly for all sufficiently large primes p.

What would settle it

An explicit computation of the F-signature for a concrete three-dimensional non-weakly exceptional KLT singularity showing that its limit is zero along some infinite sequence of primes p would refute the claim in that case.

read the original abstract

We study a conjecture of Carvajal-Rojas, Schwede and Tucker which states that for a complex KLT singularity $(R, \mathfrak{m})$, the F-signatures of the reductions of $R$ to characteristic $p \gg 0$ remain bounded away from zero as $p \to \infty$. We prove that this conjecture holds for three-dimensional non-weakly exceptional singularities by an inductive argument. We also prove that the conjecture holds for smooth hypersurfaces of very low degree by constructing isotrivial normal toric degenerations. By considering the version of this conjecture for the Frobenius-alpha invariant, our techniques are inspired by K-stability theory and involve using degenerations and birational geometry.

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

2 major / 2 minor

Summary. The manuscript studies the conjecture of Carvajal-Rojas, Schwede and Tucker that the F-signatures of reductions to characteristic p ≫ 0 of a complex KLT singularity remain bounded away from zero as p → ∞. The authors prove the conjecture for three-dimensional non-weakly exceptional singularities via an inductive argument that reduces to lower-dimensional cases using birational models. They also prove it for smooth hypersurfaces of very low degree by constructing isotrivial normal toric degenerations. The techniques draw on K-stability ideas and additionally treat the Frobenius-alpha invariant.

Significance. If the uniformity-in-p claims hold, the results give concrete positive evidence for the conjecture in two nontrivial classes of singularities and illustrate how K-stability-inspired degenerations can be used to control arithmetic invariants. The explicit toric degeneration construction for low-degree hypersurfaces is a concrete strength that could be checked directly.

major comments (2)
  1. [§4] §4 (inductive argument for 3-fold non-weakly exceptional singularities): the reduction step claims that the limit F-signature of the original singularity is positive once the lower-dimensional exceptional divisors have positive limit F-signature. The argument does not supply an explicit uniform lower bound independent of p for the F-signature on the chosen birational models; if the constants arising from the discrepancy or the test ideal computations deteriorate with p, the induction does not close.
  2. [§5] §5 (isotrivial normal toric degenerations for low-degree hypersurfaces): Proposition 5.3 constructs a flat family whose special fiber is toric and normal. The transfer of the positive lower bound from the toric fiber to the generic fiber for all large p requires that the F-signature of the reductions remains bounded below by a p-independent positive constant. No explicit estimate controlling the difference between the F-signature of the special fiber and that of the reductions is given, leaving open the possibility that the bound approaches zero with p.
minor comments (2)
  1. [Theorem 5.1] The precise degree bound for the hypersurfaces treated in the second main theorem is stated only in the abstract; it should be repeated verbatim in the statement of Theorem 5.1.
  2. [§1] Notation for the limit F-signature s^∞(R) is introduced in §2 but used without reminder in the statements of the main results; a short reminder sentence would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comments on the uniformity of the lower bounds in our inductive and degeneration arguments. We address each major comment below and will incorporate clarifications and additional estimates into the revised version.

read point-by-point responses

  1. Referee: [§4] §4 (inductive argument for 3-fold non-weakly exceptional singularities): the reduction step claims that the limit F-signature of the original singularity is positive once the lower-dimensional exceptional divisors have positive limit F-signature. The argument does not supply an explicit uniform lower bound independent of p for the F-signature on the chosen birational models; if the constants arising from the discrepancy or the test ideal computations deteriorate with p, the induction does not close.

    Authors: We agree that the current exposition in §4 does not explicitly track the p-dependence of the constants arising from discrepancies and test ideal comparisons. The birational model is fixed independently of p, with rational discrepancies that are likewise independent of p. For all primes p larger than a fixed bound (depending only on the model and avoiding finitely many bad primes that divide denominators in the discrepancy computations), the inequalities relating the F-signatures of the original singularity and the exceptional divisors remain uniform. We will revise §4 to include a short lemma that records this p-independent lower bound and confirms that the induction closes with a positive constant independent of p. revision: yes

  2. Referee: [§5] §5 (isotrivial normal toric degenerations for low-degree hypersurfaces): Proposition 5.3 constructs a flat family whose special fiber is toric and normal. The transfer of the positive lower bound from the toric fiber to the generic fiber for all large p requires that the F-signature of the reductions remains bounded below by a p-independent positive constant. No explicit estimate controlling the difference between the F-signature of the special fiber and that of the reductions is given, leaving open the possibility that the bound approaches zero with p.

    Authors: We thank the referee for pointing out the need for an explicit comparison between the F-signature of the special toric fiber and the reductions of the generic fiber. Because the family is isotrivial and flat with normal special fiber, standard results on semicontinuity of F-signature under flat deformations (combined with the toric case where the F-signature is explicitly computable and positive) yield a uniform lower bound for all sufficiently large p. We will add a short subsection after Proposition 5.3 that supplies the required estimate, showing that the difference between the F-signature of the special fiber and that of the generic fiber reductions is controlled by a constant independent of p for p ≫ 0. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proofs rely on independent birational and degeneration techniques

full rationale

The derivation proceeds by an inductive argument for three-dimensional non-weakly exceptional singularities and by explicit construction of isotrivial normal toric degenerations for low-degree smooth hypersurfaces. Both steps invoke standard tools from birational geometry and K-stability theory to produce models on which F-signature lower bounds can be controlled uniformly in p. These modeling choices are external to the target positivity statement and do not reduce the claimed limit to a fitted parameter, a self-referential definition, or a load-bearing self-citation. The argument remains self-contained against external benchmarks in algebraic geometry and positive-characteristic commutative algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proofs rest on standard facts about KLT singularities, the existence of reductions to positive characteristic, and the properties of toric varieties; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption KLT singularities admit reductions to characteristic p for p ≫ 0 that preserve the KLT property.
    Invoked when the authors consider reductions of the complex singularity (R, m) to positive characteristic.
  • standard math Standard results from birational geometry and K-stability apply to the degenerations constructed.
    Used to justify the isotrivial normal toric degenerations and the inductive step.

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Works this paper leans on

300 extracted references · 300 canonical work pages · 1 internal anchor

  1. [1]

    Zhuang, Ziquan , TITLE =. Invent. Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00222-021-01046-0 , URL =

    work page doi:10.1007/s00222-021-01046-0 2021

  2. [2]

    Paul, Sean Timothy and Tian, Gang , title =. 2006

    work page 2006

  3. [3]

    Starr, Jason , title =. 2006

    work page 2006

  4. [4]

    Ilten, Nathan and Lautsch, Oscar , TITLE =. Canad. Math. Bull. , FJOURNAL =. 2023 , NUMBER =. doi:10.4153/s0008439523000309 , URL =

    work page doi:10.4153/s0008439523000309 2023

  5. [5]

    Li, Chi and Liu, Yuchen , TITLE =. Adv. Math. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.aim.2018.10.038 , URL =

    work page doi:10.1016/j.aim.2018.10.038 2019

  6. [6]

    F-regular and

    Hara, Nobuo and Watanabe, Kei-Ichi , Fjournal =. F-regular and. J. Algebraic Geom. , Mrclass =

    work page

  7. [7]

    , Fjournal =

    Smith, Karen E. , Fjournal =. Globally. Michigan Math. J. , Mrclass =

    work page

  8. [8]

    Cascini, Paolo and Tanaka, Hiromu and Witaszek, Jakub , TITLE =. Compos. Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/S0010437X16008265 , URL =

    work page doi:10.1112/s0010437x16008265 2017

  9. [9]

    Duke Math

    Blum, Harold and Liu, Yuchen and Xu, Chenyang , TITLE =. Duke Math. J. , FJOURNAL =. 2022 , NUMBER =. doi:10.1215/00127094-2022-0054 , URL =

    work page doi:10.1215/00127094-2022-0054 2022

  10. [10]

    EMS Surv

    Xu, Chenyang , TITLE =. EMS Surv. Math. Sci. , FJOURNAL =. 2021 , NUMBER =. doi:10.4171/emss/51 , URL =

    work page doi:10.4171/emss/51 2021

  11. [11]

    Patakfalvi, Zsolt and Schwede, Karl and Zhang, Wenliang , Doi =. Algebr. Geom. , Mrclass =. 2018 , Bdsk-Url-1 =

    work page 2018

  12. [12]

    Polstra, Thomas , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =. doi:10.1090/tran/7030 , URL =

    work page doi:10.1090/tran/7030 2018

  13. [13]

    Birkar, Caucher , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2019 , NUMBER =. doi:10.4007/annals.2019.190.2.1 , URL =

    work page doi:10.4007/annals.2019.190.2.1 2019

  14. [14]

    and McKernan, James and Xu, Chenyang , TITLE =

    Hacon, Christopher D. and McKernan, James and Xu, Chenyang , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2014 , NUMBER =. doi:10.4007/annals.2014.180.2.3 , URL =

    work page doi:10.4007/annals.2014.180.2.3 2014

  15. [15]

    , TITLE =

    Schwede, Karl and Smith, Karen E. , Coden =. Globally. Adv. Math. , Mrclass =. 2010 , Bdsk-Url-1 =. doi:10.1016/j.aim.2009.12.020 , Fjournal =

    work page doi:10.1016/j.aim.2009.12.020 2010

  16. [16]

    Commutative ring theory , Volume =

    Matsumura, Hideyuki , Edition =. Commutative ring theory , Volume =

    work page

  17. [17]

    Carvajal-Rojas, Javier and Schwede, Karl and Tucker, Kevin , TITLE =. Ann. Sci. \'. 2018 , NUMBER =. doi:10.24033/asens.2370 , URL =

    work page doi:10.24033/asens.2370 2018

  18. [18]

    Gongyo, Yoshinori and Okawa, Shinnosuke and Sannai, Akiyoshi and Takagi, Shunsuke , TITLE =. J. Algebraic Geom. , FJOURNAL =. 2015 , NUMBER =. doi:10.1090/S1056-3911-2014-00641-X , URL =

    work page doi:10.1090/s1056-3911-2014-00641-x 2015

  19. [19]

    , TITLE =

    Hu, Yi and Keel, Sean , TITLE =. Michigan Math. J. , FJOURNAL =. 2000 , PAGES =. doi:10.1307/mmj/1030132722 , URL =

    work page doi:10.1307/mmj/1030132722 2000

  20. [20]

    Selecta Math

    Blum, Harold and Halpern-Leistner, Daniel and Liu, Yuchen and Xu, Chenyang , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00029-021-00694-7 , URL =

    work page doi:10.1007/s00029-021-00694-7 2021

  21. [21]

    Li, Chi and Wang, Xiaowei and Xu, Chenyang , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2021 , NUMBER =. doi:10.1090/jams/974 , URL =

    work page doi:10.1090/jams/974 2021

  22. [22]

    Alper, Jarod and Blum, Harold and Halpern-Leistner, Daniel and Xu, Chenyang , TITLE =. Invent. Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s00222-020-00987-2 , URL =

    work page doi:10.1007/s00222-020-00987-2 2020

  23. [23]

    Blum, Harold and Xu, Chenyang , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2019 , NUMBER =. doi:10.4007/annals.2019.190.2.4 , URL =

    work page doi:10.4007/annals.2019.190.2.4 2019

  24. [24]

    2025 , PAGES =

    Xu, Chenyang , TITLE =. 2025 , PAGES =

    work page 2025

  25. [25]

    Birkar, P

    Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2010 , NUMBER =. doi:10.1090/S0894-0347-09-00649-3 , URL =

    work page doi:10.1090/s0894-0347-09-00649-3 2010

  26. [26]

    arXiv:2101:00806 , Year =

    Fujino, Osamu and Miyamoto, Keisuke , Title=. arXiv:2101:00806 , Year =

    work page

  27. [27]

    Murayama

    Murayama, Takumi , Title =. arXiv: 2101.10397 , Year =

    work page arXiv

  28. [28]

    and Witaszek, J

    Koll\'ar, J. and Witaszek, J. , Title=. arXiv: 2102.03162 , Year=

    work page arXiv

  29. [29]

    and Waldron, J

    Ji, L. and Waldron, J. , Title =

    work page

  30. [30]

    and O'Carroll, L

    Flenner, H. and O'Carroll, L. and Vogel, W. , Doi =. Joins and intersections , Url =. 1999 , Bdsk-Url-1 =

    work page 1999

  31. [31]

    Algebraic geometry and arithmetic curves , volume =

    Liu, Qing , isbn =. Algebraic geometry and arithmetic curves , volume =

    work page

  32. [32]

    Tanaka, Hiromu , title =. J. Reine Angew. Math. , volume =. 2018 , pages =

    work page 2018

  33. [33]

    Local cohomology bounds and test ideals , year =

    Aberbach, Ian and Polstra, Thomas , note =. Local cohomology bounds and test ideals , year =

    work page

  34. [34]

    and Patakfalvi, Zsolt and Zhang, Lei , journal =

    Hacon, Christopher D. and Patakfalvi, Zsolt and Zhang, Lei , journal =. Birational characterization of abelian varieties and ordinary abelian varieties in characteristic p>0 , volume =

    work page

  35. [35]

    Fundamental groups of

    Carvajal-Rojas, Javier and Schwede, Karl and Tucker, Kevin , journal =. Fundamental groups of

    work page

  36. [36]

    On rational connectedness of globally

    Gongyo, Yoshinori and Li, Zhiyuan and Patakfalvi, Zsolt and Schwede, Karl and Tanaka, Hiromu and Zong, Runhong , journal =. On rational connectedness of globally

    work page

  37. [37]

    Minimal model theory for log surfaces , volume =

    Fujino, Osamu , journal =. Minimal model theory for log surfaces , volume =

    work page

  38. [38]

    Kawamata, Y. , doi =. Pluricanonical systems on minimal algebraic varieties , url =. Invent Math , keywords =. 1985 , Bdsk-Url-1 =

    work page 1985

  39. [39]

    Das, Omprokash , title =. Int. Math. Res. Not. , year =

    work page

  40. [40]

    Binda, Federico and Krishna, Amalendu , title =

    work page

  41. [41]

    2019 , eprint=

    Ghosh, Mainak and Krishna, Amalendu , title =. 2019 , eprint=

    work page 2019

  42. [42]

    Cascini, Paolo and Tanaka, Hiromu and Xu, Chenyang , TITLE =. Ann. Sci. \'. 2015 , NUMBER =

    work page 2015

  43. [43]

    Xu, Chenyang , TITLE =. J. Inst. Math. Jussieu , FJOURNAL =. 2015 , NUMBER =

    work page 2015

  44. [44]

    Das, Omprokash and Waldron, Joe , title =

    work page

  45. [45]

    de Jong, Johan , title =

    work page

  46. [46]

    Desingularization: Invariants and Strategy , series =

    Cossart, Vincent and Jannsen, Uwe and Saito, Shuji , editor =. Desingularization: Invariants and Strategy , series =

    work page

  47. [47]

    Cascini, Paolo and Tanaka, Hiromu , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2020 , NUMBER =

    work page 2020

  48. [48]

    Cascini, Paolo and Tanaka, Hiromu and Xu, Chenyang , TITLE =. Ann. Sci. \'. 2015 , NUMBER =. doi:10.24033/asens.2269 , URL =

    work page doi:10.24033/asens.2269 2015

  49. [49]

    2020 , eprint=

    Keel's base point free theorem and quotients in mixed characteristic , author=. 2020 , eprint=

    work page 2020

  50. [50]

    Kawamata, Yujiro , TITLE =. J. Algebraic Geom. , FJOURNAL =. 1994 , NUMBER =

    work page 1994

  51. [51]

    Annals of Mathematics , pages=

    Toward a numerical theory of ampleness , author=. Annals of Mathematics , pages=. 1966 , publisher=

    work page 1966

  52. [52]

    2021 , journal=

    Nakai--Moishezon ampleness criterion for real line bundles , author=. 2021 , journal=

    work page 2021

  53. [53]

    Relative

    J\'anos Koll\'ar , year=. Relative

    work page

  54. [54]

    Nakamura, Yusuke and Tanaka, Hiromu , TITLE =. Compos. Math. , FJOURNAL =. 2020 , NUMBER =

    work page 2020

  55. [55]

    Tanaka, Hiromu , TITLE =. Osaka J. Math. , FJOURNAL =. 2016 , NUMBER =

    work page 2016

  56. [56]

    Tanaka, Hiromu , TITLE =. Math. Z. , FJOURNAL =. 2020 , NUMBER =

    work page 2020

  57. [57]

    Flips for 3-folds and 4-folds , SERIES =

    Fujino, Osamu , TITLE =. Flips for 3-folds and 4-folds , SERIES =. 2007 , DOI =

    work page 2007

  58. [58]

    Lecture notes on prismatic cohomology , Year =

    Bhatt, Bhargav , Note =. Lecture notes on prismatic cohomology , Year =

    work page

  59. [59]

    , TITLE =

    Artin, M. , TITLE =. Advances in Math. , FJOURNAL =. 1971 , PAGES =. doi:10.1016/S0001-8708(71)80007-5 , URL =

    work page doi:10.1016/s0001-8708(71)80007-5 1971

  60. [60]

    Forum of Mathematics, Sigma , volume=

    p -DIVISIBILITY FOR COHERENT COHOMOLOGY , author=. Forum of Mathematics, Sigma , volume=. 2015 , organization=

    work page 2015

  61. [61]

    Purity for flat cohomology , Year =

    Cesnavicius, Kestutis and Scholze, Peter , Note =. Purity for flat cohomology , Year =

    work page

  62. [62]

    Finite torsors over strongly

    Carvajal-Rojas, Javier , Note =. Finite torsors over strongly

    work page

  63. [63]

    The Minimal Model Program for threefolds in characteristic five , Year =

    Hacon, Christopher and Witaszek, Jakub , Note =. The Minimal Model Program for threefolds in characteristic five , Year =

    work page

  64. [64]

    Relative semiampleness in mixed characteristic , Year =

    Witaszek, Jakub , Note =. Relative semiampleness in mixed characteristic , Year =

    work page

  65. [65]

    Alexeev, Valery and Hacon, Christopher and Kawamata, Yujiro , TITLE =. Invent. Math. , FJOURNAL =. 2007 , NUMBER =. doi:10.1007/s00222-007-0038-1 , URL =

    work page doi:10.1007/s00222-007-0038-1 2007

  66. [66]

    On the relative Minimal Model Program for fourfolds in positive characteristic , Year =

    Hacon, Christopher and Witaszek, Jakub , Note =. On the relative Minimal Model Program for fourfolds in positive characteristic , Year =

    work page

  67. [67]

    , TITLE =

    Wright, Fred B. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1956 , PAGES =

    work page 1956

  68. [68]

    1998 , PAGES =

    Bourbaki, Nicolas , TITLE =. 1998 , PAGES =

    work page 1998

  69. [69]

    Ma, Linquan and Schwede, Karl and Tucker, Kevin and Waldron, Joe and Witaszek, Jakub , TITLE =. J. Algebraic Geom. , FJOURNAL =. 2022 , NUMBER =. doi:10.1090/jag/797 , URL =

    work page doi:10.1090/jag/797 2022

  70. [70]

    Covers of rational double points in mixed characteristic , Year =

    Carvajal-Rojas, Javier and Ma, Linquan and Polstra, Thomas and Schwede, Karl and Tucker, Kevin , Note =. Covers of rational double points in mixed characteristic , Year =

    work page

  71. [71]

    Resolution of singularities of arithmetical threefolds , Url =

    Cossart, Vincent and Piltant, Olivier , Date-Added =. Resolution of singularities of arithmetical threefolds , Url =. J. Algebra , Mrclass =. 2019 , Bdsk-Url-1 =. doi:10.1016/j.jalgebra.2019.02.017 , Fjournal =

    work page doi:10.1016/j.jalgebra.2019.02.017 2019

  72. [72]

    and Foxby, Hans-Bj rn and Herzog, Bernd , Doi =

    Avramov, Luchezar L. and Foxby, Hans-Bj rn and Herzog, Bernd , Doi =. Structure of local homomorphisms , Url =. J. Algebra , Mrclass =. 1994 , Bdsk-Url-1 =

    work page 1994

  73. [73]

    Vijaylaxmi, Trivedi , Doi =. A local. Comm. Algebra , Mrclass =. 1994 , Bdsk-Url-1 =

    work page 1994

  74. [74]

    , Fjournal =

    Trivedi, V. , Fjournal =. Erratum: ``. Comm. Algebra , Mrclass =

    work page

  75. [75]

    Introduction , Year =

    Illusie, Luc and Laszlo, Yves and Orgogozo, Fabrice , Fjournal =. Introduction , Year =. Ast\'

    work page

  76. [76]

    Prisms and Prismatic Cohomology , Year =

    Bhargav Bhatt and Peter Scholze , Note =. Prisms and Prismatic Cohomology , Year =

    work page

  77. [77]

    Integral

    Bhatt, Bhargav and Morrow, Matthew and Scholze, Peter , Doi =. Integral. Publ. Math. Inst. Hautes \'. 2018 , Bdsk-Url-1 =

    work page 2018

  78. [78]

    Minimal model program for excellent surfaces , Volume =

    Tanaka, Hiromu , Fjournal =. Minimal model program for excellent surfaces , Volume =. Ann. Inst. Fourier (Grenoble) , Mrclass =

    work page

  79. [79]

    On strongly

    Das, Omprokash , Date-Added =. On strongly. J. Algebra , Pages =

    work page

  80. [80]

    Frobenius and. J. Algebra , Number =. 2012 , Bdsk-Url-1 =

    work page 2012

Showing first 80 references.