Pith. sign in

REVIEW 3 major objections 4 minor 2 cited by

On Grauert-Riemenschneider vanishing for Cohen-Macaulay schemes of klt type

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that Cohen-Macaulay schemes of klt type satisfy the degree-one Grauert-Riemenschneider vanishing $R^1\pi_*\omega_Y=0$, and that three-dimensional such schemes have rational singularities.

desk verdict A natural result with a real gap in the key descent claim; the main theorem is unproved as written but the paper deserves referee attention. read the letter →

arxiv 2506.21381 v1 pith:S2JXECAN submitted 2025-06-26 math.AG

Jefferson Baudin , Tatsuro Kawakami , Linus Rösler This is my paper · ORCID
classification math.AG MSC 14F1714B0513A35
keywords Grauert-RiemenschneidervanishingkltsingularitiesCohen-MacaulayschemesrationalpositivecharacteristicWittvectorsQ_p-rationalhigherdirectimages
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

The paper asks whether Grauert-Riemenschneider vanishing, which is known to fail in positive characteristic, survives for singularities that are both of klt type and Cohen-Macaulay. It proves the first missing case: for any resolution $\pi:Y\to X$ of such a scheme, $R^1\pi_*\omega_Y=0$. Since the top-degree vanishing $R^{d-1}\pi_*\omega_Y=0$ was already known, a three-dimensional Cohen-Macaulay klt-type scheme now satisfies full Grauert-Riemenschneider vanishing and has rational singularities. The same argument yields $\mathbb{Q}_p$-rational singularities for Cohen-Macaulay klt-type schemes of any dimension over a perfect field of characteristic $p>0$.

What carries the argument

The engine is a codimension bound for higher direct images of the structure sheaf (Theorem 3.2): for a klt-type variety $X$ and resolution $\pi:Y\to X$, every $i>0$ satisfies $\operatorname{codim} \operatorname{Supp} R^i\pi_*\mathcal{O}_Y > i+1$. The proof proceeds by induction on dimension, subtracting exceptional divisors $F_j$ one at a time through short exact sequences; the choice of $F_j$ is governed by klt type, which supplies coefficients $a_i<1$ so that $-(K_Y+\sum n_iF_i)$ restricts to a big line bundle on $F_j$. Lemma 3.4 then converts this codimension estimate into the desired vanishing: Grothendieck duality and the Cohen-Macaulay property place $R^1\pi_*\omega_Y$ inside $H^{-(d-1)}(\omega_X^\bullet)$, which vanishes.

What would settle it

Construct a Cohen-Macaulay klt-type threefold over a field of characteristic 2, 3, or 5 with a resolution $\pi:Y\to X$ for which $R^1\pi_*\omega_Y\neq0$; that would directly contradict Corollary 1.4. Concretely, take one of the known non-Cohen-Macaulay klt counterexamples to Grauert-Riemenschneider vanishing and test whether a small Cohen-Macaulay modification remains of klt type and still has a nonzero first direct image.

Watch

Extended reading notes

Core claim

The central result is Theorem 1.2(2): if $X$ is a Noetherian excellent normal scheme of klt type, meaning there is an effective $\mathbb{Q}$-divisor $\Delta$ making $(X,\Delta)$ klt, and $X$ is also Cohen-Macaulay, then every resolution $\pi:Y\to X$ satisfies $R^1\pi_*\omega_Y=0$. Together with the known vanishing $R^{d-1}\pi_*\omega_Y=0$, this gives Corollary 1.4 in dimension three: $R\pi_*\omega_Y=\omega_X$, so $X$ satisfies Grauert-Riemenschneider vanishing and has rational singularities. In positive characteristic, the paper deduces Theorem 1.5: a Cohen-Macaulay klt-type scheme of finite type over a perfect field is $\mathbb{Q}_p$-rational, and with projectivity plus isolated singularities it is Witt-rational. The authors note that the Cohen-Macaulay hypothesis cannot simply be dropped, because the known positive-characteristic counterexamples to Grauert-Riemenschneider vanishing are all non-Cohen-Macaulay.

Load-bearing premise

The load-bearing premise is that the non-normal exceptional divisors in the induction still satisfy the duality and bigness statements used to conclude vanishing, a step the authors flag but do not spell out, and for the $\mathbb{Q}_p$-rationality half the companion paper's theorem must hold.

Editorial extensions

If this is right

  • Three-dimensional Cohen-Macaulay klt-type schemes satisfy full Grauert-Riemenschneider vanishing, so $R\pi_*\omega_Y=\omega_X$ for every resolution and they have rational singularities.
  • Strongly F-regular and quasi-F-regular threefolds in positive characteristic satisfy Grauert-Riemenschneider vanishing and have rational singularities, since such varieties are Cohen-Macaulay and of klt type.
  • Globally +-regular three-dimensional pairs with $\mathbb{Q}$-Cartier $K_X+\Delta$ also satisfy Grauert-Riemenschneider vanishing and have rational singularities.
  • In arbitrary dimension over a perfect field of characteristic $p>0$, a Cohen-Macaulay klt-type scheme is $\mathbb{Q}_p$-rational; adding projectivity and isolated singularities upgrades this to Witt-rationality.
  • The degree-one vanishing $R^1\pi_*\omega_Y=0$ holds in every dimension for Cohen-Macaulay klt-type schemes, with $\pi_*\omega_Y=\omega_X$ as part of the proof.

Reading between the lines

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

  • I infer that Cohen-Macaulayness, rather than characteristic zero, is the structural condition protecting degree-one Grauert-Riemenschneider vanishing; a natural next test is whether the same induction pushes the vanishing to all higher degrees $R^i\pi_*\omega_Y=0$ for Cohen-Macaulay klt-type schemes of any dimension.
  • Because the proof uses only a resolution and not a log resolution, I infer it may adapt to settings where log resolutions are unavailable, such as mixed characteristic or low-dimensional positive characteristic.
  • The $\mathbb{Q}_p$-rationality result is stated under Cohen-Macaulayness, but the authors note it only needs $\mathbb{Q}_p$-Cohen-Macaulayness, a Frobenius-theoretic depth condition preserved under universal homeomorphisms and finite quotients; I infer this makes the result applicable to quotient singularities where ordinary Cohen-Macaulayness fails.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies Grauert-Riemenschneider (GR) vanishing for Cohen-Macaulay schemes of klt type. The main theorem asserts that if X is a Noetherian excellent normal scheme of klt type and Cohen-Macaulay, and π:Y→X is a resolution, then R^1π_*ω_Y=0. The authors combine this with a stated general vanishing R^{d-1}π_*ω_Y=0 to deduce that three-dimensional CM klt-type schemes satisfy full GR vanishing and have rational singularities. A further theorem asserts Q_p-rationality for klt-type schemes over perfect fields of characteristic p>0, using a companion paper by the first author.

Significance. If correct, the paper would answer a natural question raised by the known counterexamples to GR vanishing in positive characteristic: all known counterexamples are non-Cohen-Macaulay, and the paper would show that the CM assumption is enough to restore vanishing in degree one, with a clean three-dimensional corollary. The strategy via codimension bounds on R^iπ_*O_Y and Grothendieck duality is attractive and, if made sound, would be a useful contribution. However, the current proof has two load-bearing gaps: the descent Claim in Theorem 3.2 can select an exceptional divisor whose coefficient is already zero, and Proposition 3.1 is false as stated. These issues affect the main theorem and the advertised corollary respectively.

major comments (3)
  1. [Theorem 3.2, Claim (page 5)] The descent Claim is not proved. In the Claim, J is defined as {i | n_i−a_i>0}, and the chosen j is a maximizer of (n_i−a_i)/r_i. Nothing forces n_j≥1, because a_i can be negative: with the paper's convention K_Y+Σa_iF_i∼_Qπ^*(K_X+Δ), for a blow-up of a smooth variety along a codimension-c center one has a_i=1−c, which is negative for c≥2. Thus a divisor with n_i=0 can satisfy n_i−a_i=−a_i>0 and can attain the maximum. Concretely, for the blow-up of a smooth 5-fold along the disjoint union of a point and a surface, with Δ=0, the two exceptional components have a_1=−4 and a_2=−2, and r_1=r_2=n. At coefficients (n_1,n_2)=(0,1), the ratios are 4/n and 3/n, so the proof selects j=1 with n_1=0. The conclusion would assert vanishing for O_Y(F_1−F_2), not a reduction toward O_Y. The iterative descent therefore stalls, and Theorem 3.2, and hence Theorem 1.2(2), is not established by the given argument.
  2. [Proposition 3.1 (page 3)] Proposition 3.1 is false as stated. It claims that R^{d-1}π_*ω_Y=0 for every resolution of every d-dimensional variety. For d=2 this would imply every normal surface singularity is rational, since for a resolution of a surface, R^1π_*ω_Y is the local-duality counterpart of R^1π_*O_Y. A cone over an elliptic curve is a normal surface singularity with nonzero R^1π_*O_Y and hence nonzero R^1π_*ω_Y. The proof's reduction to dim 2 and citation of [Kol13, Theorem 10.4] appears to import a statement that does not hold without additional hypotheses. Moreover, the proof invokes 'relative Serre vanishing' to assert R^{d-1}π_*ω_X(H)=0 for a single hyperplane twist; relative Serre vanishing gives vanishing for sufficiently high powers of a relatively ample line bundle, not for the first twist. Since Corollary 1.4 and Remark 1.3(a) rely on Proposition 3.1, this needs to be corrected: either the proposition must be restricted to an appropriate class (e.g. klt type, with a valid proof), or Corollary 1.4 must be proved by a different argument that supplies R^{d-1}π_*ω_Y=0 for the CM klt threefold case.
  3. [Theorem 3.2, Claim, non-normal F_j (Remark 3.3)] The Claim's bigness and duality steps are applied to exceptional divisors F_j that are only integral, not necessarily normal. The displayed Serre-duality isomorphism for F_j and the restriction argument for bigness need a justification in this non-normal setting. The authors state in Remark 3.3 that they work with Q-line bundles, but the proof as written uses a Serre-duality statement involving K_{F_j} for a possibly non-normal divisor. Since F_j is a Cartier divisor on a regular scheme Y, it is Gorenstein and Cohen-Macaulay, so a version of Serre duality may be available, but this is not explained. This issue is less severe than the coefficient-descent gap, but it is load-bearing for the Claim and should be addressed in a revision.
minor comments (4)
  1. [Proposition 3.1 proof (page 3)] There is a notational slip: the proof writes R^{d-1}π_*ω_X(H), but the sheaf should live on Y, so it should be R^{d-1}π_*(ω_Y⊗π^*O_X(H)) or similar.
  2. [Theorem 3.2 proof (page 5)] The condition a_i∈Q_{<1} is nonstandard and confusing. For a klt pair written as K_Y+Σa_iF_i+π_*^{-1}Δ∼_Qπ^*(K_X+Δ), the discrepancies normally satisfy a_i>−1. Please clarify the sign convention and the relation to the klt assumption.
  3. [Throughout] There are several small typos, e.g. 'we can reduced' should be 'we can reduce', and the displayed exact sequence after Proposition 3.1 uses ω_X(H) where the intended sheaf is on Y; these should be corrected.
  4. [Theorem 1.5 and Remark 1.6] The proof of Theorem 1.5 depends on [Bau25, Theorem 5.1.4], an unpublished companion paper by the first author. This dependency should be explicitly flagged as a preprint dependency, and the referee should be able to check that theorem if the publication decision relies on it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main vanishing theorem is derived from standard inputs, and the Qp-rationality statement rests on an external companion theorem rather than an input-output equivalence.

full rationale

The derivation of the main vanishing theorem (Theorem 1.2) is self-contained within the paper's stated framework: Theorem 3.2 is built from the klt discrepancy formula, relative Serre vanishing, the π-ampleness of OY(−E) on the blow-up, and Serre duality on the exceptional divisor; Lemma 3.4 and Proposition 3.1 use standard Grothendieck duality and known low-dimensional vanishing. No parameter is fitted, and no conclusion is identified with its hypothesis by construction. The only result imported from the authors' own prior work is the Qp-rationality criterion [Bau25, Theorem 5.1.4] in Theorem 1.5; that is cited as an external theorem with stated assumptions rather than as a restatement of the present paper's vanishing, and it does not feed back into Theorem 1.2 or Corollary 1.4. The issues flagged in Remark 3.3 and in the skeptical analysis (non-normal exceptional divisors, and the possible selection of an index with n_j = 0 in the Claim) are proof-completeness or correctness concerns, not circularity, because they do not make the conclusions equivalent to the inputs by definition.

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

No free parameters are fitted; the paper is a proof-based contribution. The main novel content is Theorem 1.2, which rests on standard machinery plus the klt type hypothesis. The arbitrary-dimensional Q_p-rationality result inherits the burden of the companion paper [Bau25].

assumptions (5)
  • standard math Grothendieck duality for π and the functor RHom(-, ω_X^•)
    Invoked in Lemma 3.4 to identify D(Rπ_*O_Y) and derive the vanishing.
  • standard math Relative Serre vanishing for projective morphisms
    Used in Theorem 3.2 and Proposition 3.1 to kill R^{d-1} of O_Y(-nE) and ω_Y(H).
  • domain assumption Existence of a boundary Δ with (X, Δ) klt, and the discrepancy formula K_Y + Σ a_i F_i + π_*^{-1}Δ ~_Q π^*(K_X + Δ) with a_i < 1
    Central to the bigness argument in the Claim of Theorem 3.2; it uses the klt type hypothesis.
  • domain assumption Serre duality and the bigness/anti-effectiveness principle on the possibly non-normal prime divisors F_j
    Needed in the Claim to conclude H^0(F_j, O_{F_j}(K_Y + Σ n_i F_i)) = 0 from the bigness of its negative; the paper notes the non-normality issue in Remark 3.3 but does not give full details.
  • domain assumption [Bau25, Theorem 5.1.4]
    Used as a black box to deduce Theorem 1.5 from Lemma 3.8; the companion paper is by the first author and is not verified here.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On Grauert-Riemenschneider vanishing for Cohen-Macaulay schemes of klt type." pith.science (2026). https://pith.science/paper/S2JXECAN

@misc{pith2026250621381,
  author       = {Pith},
  title        = {Pith review of: On Grauert-Riemenschneider vanishing for Cohen-Macaulay schemes of klt type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S2JXECAN}},
  note         = {Machine review of arXiv:2506.21381}
}
abstract

Given a Cohen-Macaulay scheme of klt type $X$ and a resolution $\pi\colon Y\to X$, we show that $R^1\pi_*\omega_Y=0$. We deduce that if $\mathrm{dim}(X)=3$, then $X$ satisfies Grauert-Riemenschneider vanishing and therefore has rational singularities. We also obtain that in arbitrary dimension, if $X$ is of finite type over a perfect field of characteristic $p>0$, then $X$ has $\mathbb{Q}_p$-rational singularities.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the Euler characteristic of weakly ordinary varieties of maximal Albanese dimension

    math.AG 2025-07 conditional novelty 7.0 of 10

    Smooth proper weakly ordinary varieties of maximal Albanese dimension satisfy chi(X, omega_X) >= 0, with chi = 0 for non-general-type examples and the Albanese image then fibered by ordinary abelian varieties.

  2. On Steenbrink vanishing for rational singularities in positive characteristic

    math.AG 2025-07 conditional novelty 6.0 of 10

    A positive characteristic Steenbrink vanishing theorem is proved for rational singularities, giving the vanishing for strongly F-regular threefolds and Q-factorial klt threefolds in large characteristic.

Reference graph

Works this paper leans on

37 extracted references · 17 canonical work pages · cited by 2 Pith papers

  1. [1]

    On the K awamata- V iehweg vanishing theorem for log del P ezzo surfaces in positive characteristic

    Emelie Arvidsson, Fabio Bernasconi, and Justin Lacini. On the K awamata- V iehweg vanishing theorem for log del P ezzo surfaces in positive characteristic. Compos. Math. , 158(4):750--763, 2022

    2022

  2. [2]

    A G rauert-- R iemenschneider vanishing theorem for W itt canonical sheaves

    Jefferson Baudin. A G rauert-- R iemenschneider vanishing theorem for W itt canonical sheaves. arXiv e-print: arXiv:2506.14647v1 , 2025. Available at arXiv:2506.14647 https://arxiv.org/abs/2506.14647

    arXiv 2025

  3. [3]

    Spectrum of differential operators with elliptic adjoint on a scale of localized Sobolev spaces

    Jefferson Baudin, Fabio Bernasconi, and Tatsuro Kawakami. The Frobenius--stable version of the Grauert--Riemenschneider vanishing theorem fails . 2023. Available at arXiv:2102.13456v3 https://arxiv.org/abs/2312.13456

    work page Pith review arXiv 2023

  4. [4]

    Rational singularities and rational points

    Manuel Blickle and H \'e l \`e ne Esnault. Rational singularities and rational points. Pure Appl. Math. Q. , 4(3, part 2):729--741, 2008

    work page 2008

  5. [5]

    Kawamata- V iehweg vanishing fails for log del P ezzo surfaces in characteristic 3

    Fabio Bernasconi. Kawamata- V iehweg vanishing fails for log del P ezzo surfaces in characteristic 3. J. Pure Appl. Algebra , 225(11):Paper No. 106727, 16, 2021

    work page 2021

  6. [6]

    Vanishing theorems for three-folds in characteristic p>5

    Fabio Bernasconi and J\'anos Koll\'ar. Vanishing theorems for three-folds in characteristic p>5 . Int. Math. Res. Not. IMRN , (4):2846--2866, 2023

    2023

  7. [7]

    Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic

    Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek. Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic. Publ. Math. Inst. Hautes \'Etudes Sci. , 138:69--227, 2023

    work page 2023

  8. [8]

    Higher direct images of the structure sheaf in positive characteristic

    Andre Chatzistamatiou and Kay R\" u lling. Higher direct images of the structure sheaf in positive characteristic. Algebra Number Theory , 5(6):693--775, 2011

    2011

Show all 37 references
  1. [9]

    Hodge- W itt cohomology and W itt-rational singularities

    Andre Chatzistamatiou and Kay R\" u lling. Hodge- W itt cohomology and W itt-rational singularities. Doc. Math. , 17:663--781, 2012

    2012

  2. [10]

    Vanishing of the higher direct images of the structure sheaf

    Andre Chatzistamatiou and Kay R\"ulling. Vanishing of the higher direct images of the structure sheaf. Compos. Math. , 151(11):2131--2144, 2015

    2015

  3. [11]

    Purely log terminal threefolds with non-normal centres in characteristic two

    Paolo Cascini and Hiromu Tanaka. Purely log terminal threefolds with non-normal centres in characteristic two. Amer. J. Math. , 141(4):941--979, 2019

    2019

  4. [12]

    Singularit\' e s rationnelles et d\' e formations

    Ren\' e e Elkik. Singularit\' e s rationnelles et d\' e formations. Invent. Math. , 47(2):139--147, 1978

    1978

  5. [13]

    Rationalit\' e des singularit\' e s canoniques

    Ren\' e e Elkik. Rationalit\' e des singularit\' e s canoniques. Invent. Math. , 64(1):1--6, 1981

    1981

  6. [14]

    On the depth of local rings of invariants of cyclic groups

    John Fogarty. On the depth of local rings of invariants of cyclic groups. Proc. Amer. Math. Soc. , 83(3):448--452, 1981

    1981

  7. [15]

    Notes on some t -structures

    Ofer Gabber. Notes on some t -structures. In Geometric aspects of D work theory. V ol. I , II , pages 711--734. Walter de Gruyter, Berlin, 2004

    2004

  8. [16]

    Rational points on log F ano threefolds over a finite field

    Yoshinori Gongyo, Yusuke Nakamura, and Hiromu Tanaka. Rational points on log F ano threefolds over a finite field. J. Eur. Math. Soc. (JEMS) , 21(12):3759--3795, 2019

    2019

  9. [17]

    Residues and duality

    Robin Hartshorne. Residues and duality . Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin, 1966

    1963

  10. [18]

    Algebraic geometry

    Robin Hartshorne. Algebraic geometry . Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52

    1977

  11. [19]

    Tight closure and strong F -regularity

    Melvin Hochster and Craig Huneke. Tight closure and strong F -regularity. In Colloque en l'honneur de Pierre Samuel (Orsay, 1987) , number 38 in M\' e m. Soc. Math. France (N.S.), pages 119--133. Soci\'et\'e math\'ematique de France, 1989

    1987

  12. [20]

    Hacon and S\' a ndor J

    Christopher D. Hacon and S\' a ndor J. Kov\' a cs. Generic vanishing fails for singular varieties and in characteristic p>0 . In Recent advances in algebraic geometry , volume 417 of London Math. Soc. Lecture Note Ser. , pages 240--253. Cambridge Univ. Press, Cambridge, 2015

    2015

  13. [21]

    F-regular and F -pure rings vs

    Nobuo Hara and Kei-Ichi Watanabe. F-regular and F -pure rings vs. log terminal and log canonical singularities. J. Algebraic Geom. , 11(2):363--392, 2002

    2002

  14. [22]

    Hacon and Jakub Witaszek

    Christopher D. Hacon and Jakub Witaszek. On the rationality of K awamata log terminal singularities in positive characteristic. Algebr. Geom. , 6(5):516--529, 2019

    2019

  15. [23]

    Hacon and Jakub Witaszek

    Christopher D. Hacon and Jakub Witaszek. On the relative minimal model program for threefolds in low characteristics. Peking Math. J. , 5(2):365--382, 2022

    2022

  16. [24]

    Hacon and Jakub Witaszek

    Christopher D. Hacon and Jakub Witaszek. On the relative minimal model program for fourfolds in positive and mixed characteristic. Forum Math. Pi , 11:Paper No. e10, 35, 2023

    2023

  17. [25]

    On vanishing of higher direct images of the structure sheaf

    Shihoko Ishii and Ken-Ichi Yoshida. On vanishing of higher direct images of the structure sheaf. arXiv e-print, arXiv:2410.15282v3 , 2024. Available at arXiv:2410.15282 https://arxiv.org/abs/2410.15282

    2024 arXiv

  18. [26]

    Singularities of the minimal model program , volume 200 of Cambridge Tracts in Mathematics

    J\' a nos Koll\' a r. Singularities of the minimal model program , volume 200 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2013. With a collaboration of S\' a ndor Kov\' a cs

    2013

  19. [27]

    Kov\' a cs

    S\' a ndor J. Kov\' a cs. A characterization of rational singularities. Duke Math. J. , 102(2):187--191, 2000

    2000

  20. [28]

    Quasi- F -splittings in birational geometry

    Tatsuro Kawakami, Teppei Takamatsu, Hiromu Tanaka, Jakub Witaszek, Fuetaro Yobuko, and Shou Yoshikawa. Quasi- F -splittings in birational geometry. https://arxiv.org/abs/2208.08016, 2022. To appear in Ann. Sci. \'Ec. Norm. Sup\'er. (4)

    2022 arXiv

  21. [29]

    Quasi- F -splittings in birational geometry III

    Tatsuro Kawakami, Teppei Takamatsu, Hiromu Tanaka, Jakub Witaszek, Fuetaro Yobuko, and Shou Yoshikawa. Quasi- F -splittings in birational geometry III . https://arxiv.org/abs/2408.01921, 2024

    2024

  22. [30]

    Ordinary varieties with trivial canonical bundle are not uniruled

    Zsolt Patakfalvi and Maciej Zdanowicz. Ordinary varieties with trivial canonical bundle are not uniruled. Math. Ann. , 380(3-4):1767--1799, 2021

    2021

  23. [31]

    Contre-exemple au ``vanishing theorem'' en caract\' e ristique p>0

    Michel Raynaud. Contre-exemple au ``vanishing theorem'' en caract\' e ristique p>0 . In C. P . R amanujam---a tribute , volume 8 of Tata Inst. Fundam. Res. Stud. Math. , pages 273--278. Springer, Berlin-New York, 1978

    1978

  24. [32]

    Karl Schwede and Karen E. Smith. Globally F -regular and log F ano varieties. Adv. Math. , 224(3):863--894, 2010

    2010

  25. [33]

    The S tacks project

    The Stacks project authors . The S tacks project. https://stacks.math.columbia.edu, 2025

    2025

  26. [34]

    An interpretation of multiplier ideals via tight closure

    Shunsuke Takagi. An interpretation of multiplier ideals via tight closure. J. Algebraic Geom. , 13(2):393--415, 2004

    2004

  27. [35]

    The failure of K odaira vanishing for F ano varieties, and terminal singularities that are not C ohen- M acaulay

    Burt Totaro. The failure of K odaira vanishing for F ano varieties, and terminal singularities that are not C ohen- M acaulay. J. Algebraic Geom. , 28(4):751--771, 2019

    2019

  28. [36]

    Terminal 3 -folds that are not C ohen- M acaulay

    Burt Totaro. Terminal 3 -folds that are not C ohen- M acaulay. arXiv e-print, arXiv:2407.02608v2 , 2024. Available at arXiv:2407.02608 https://arxiv.org/abs/2407.02608

    2024 arXiv

  29. [37]

    Quasi-- F^e --splittings and quasi-- F --regularity

    Hiromu Tanaka, Jakub Witaszek, and Fuetaro Yobuko. Quasi-- F^e --splittings and quasi-- F --regularity . 2024. Available at arXiv:2404.06788 https://arxiv.org/abs/2404.06788

    2024 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.