Pith. sign in

REVIEW 2 major objections 4 minor 48 references

On volumes and the generic invariance of Fano type varieties

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read Sparse Fano type fibers whose anti-canonical volumes are constant force the geometric generic fiber to be Fano type, and the same principle works in dimension two and in reduction mod p.

desk verdict A plausible and useful constant-volume criterion for Fano type fibers, with a real but openly acknowledged gap in the (b)⇒(a) direction of Theorem 1.2. read the letter →

arxiv 2506.13603 v5 pith:R674KZ52 submitted 2025-06-16 math.AG

Donghyeon Kim This is my paper · ORCID
classification math.AG MSC 14E0514E3014J4514G17
keywords Fanotypevarietiesanti-canonicalvolumegenericinvariancelogcanonicalthresholdquasi-monomialvaluationsgloballyF-regularreductionmodpDCCproperty
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 establishes that the Fano type property, which is classically known to propagate from an open set of fibers, also propagates from a Zariski-dense but possibly sparse set of fibers, provided the anti-canonical volumes on those fibers are a fixed positive constant. It proves the analogous statement without any volume condition when the Fano fibers have dimension 2. In the reduction mod p direction, it proves the same propagation when the dense fibers are globally F-regular with constant anti-canonical volume. A reader should care because this converts pointwise boundedness data into a generic geometric conclusion, and it yields a descending chain condition for anti-canonical volumes in bounded families of Fano type surfaces.

What carries the argument

The central test is the log canonical threshold criterion: a projective pair with big anticanonical class is Fano type exactly when $\operatorname{lct}_\sigma(X,\Delta,-(K_X+\Delta))>1$, where $\operatorname{lct}_\sigma$ is the infimum of log discrepancies divided by Nakayama's asymptotic order along valuations. To make this test work across a family, the paper uses the theorem that a minimizing valuation for the log canonical threshold is quasi-monomial, so only a single log-smooth model is needed, and Jiao's theorem that the volume of a divisor on the geometric generic fiber is the infimum of its volumes over any Zariski-dense set of closed fibers. Weighted blow-ups of the log-smooth model extract the quasi-monomial valuation as a divisor simultaneously on the generic and every closed fiber, and the volume-drop criterion for asymptotic orders then lets the author compare log canonical thresholds on closed fibers with those on the generic fiber. For the surface case, the machinery is Zariski decomposition of the anticanonical divisor, specialized to fibers and used to identify a klt boundary on the generic fiber.

What would settle it

Take the family constructed in the proof of (b)$\Rightarrow$(a) over an algebraically closed field of characteristic zero that is not C, and check on one closed fiber whether the restricted contractions form an MMP and whether the anti-canonical volume equals the constant v; a single fiber where the volume differs or the contractions are not an MMP would refute Theorem 1.2 in the stated generality.

Watch

Extended reading notes

Core claim

The paper's main equivalence is Theorem 1.2: for a projective surjective morphism $(X,\Delta)\to S$ between normal varieties over an algebraically closed field of characteristic zero, condition (a)---there is a Zariski-dense set $S'\subseteq S$ and a positive constant $v$ such that every fiber $(X_s,\Delta_s)$ for $s\in S'$ is Fano type and $\operatorname{vol}(-(K_{X_s}+\Delta_s))=v$---is equivalent to condition (b), namely that the geometric generic fiber $(X_\eta,\Delta_\eta)$ is Fano type. The direction (a)$\Rightarrow$(b) is proved by showing that a log canonical threshold criterion for Fano type pairs, stated as $\operatorname{lct}_\sigma>1$, passes from the dense fibers to the generic fiber. The reverse direction shows that after shrinking the base, all fibers are Fano type with anti-canonical volume $v$, by running an MMP on the generic fiber and restricting its steps to fibers. The paper also proves the dimension-2 case of the conjectured generic invariance without a volume hypothesis, and a reduction-mod-p analogue in which the role of Fano type is played by global F-regularity.

Load-bearing premise

The load-bearing premise is that a fixed sequence of birational contraction steps (an MMP) run on the generic fiber, when restricted to each closed fiber, is still a valid MMP there and preserves both the Fano type property and the anti-canonical volume; the paper states this is anticipated and cites a proof only over C, not over arbitrary algebraically closed fields of characteristic zero.

Editorial extensions

If this is right

  • If Theorem 1.2 stands, then a family whose Fano type fibers form any Zariski-dense subset, with all anti-canonical volumes equal to the same positive number, has a Fano type geometric generic fiber; by the reverse direction, the same volume then propagates to a neighborhood of the base.
  • Fano type surfaces are generically invariant with no volume hypothesis: Conjecture 1.1 holds in relative dimension two.
  • Anti-canonical volumes over a bounded family of Fano type surfaces satisfy the descending chain condition, so strictly decreasing infinite sequences of such volumes are impossible.
  • In the reduction mod p setting, constant anti-canonical volume on a Zariski-dense set of globally F-regular fibers forces the characteristic-zero limit pair to be Fano type.
  • The log canonical threshold criterion used here provides a practical fiberwise test for Fano type that needs only one log resolution, not all of them.

Reading between the lines

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

  • The volume-constancy hypothesis may be relaxable to a DCC condition on the set of volumes: Corollary 1.4 already extracts constancy along a subsequence from boundedness, and the same mechanism may work in higher dimensions under boundedness assumptions.
  • The gap between the theorem and its stated generality is a relative MMP statement; proving that restricted D-MMP sequences specialize to D-MMPs over arbitrary algebraically closed fields of characteristic zero would complete the (b)$\Rightarrow$(a) direction.
  • The same $\operatorname{lct}_\sigma$ comparison could test the mod p conjecture directly: a globally F-regular type variety with a model whose mod p fibers have constant anti-canonical volumes should be Fano type, using Theorem 1.6's method without further hypotheses.
  • For surfaces, the proof treats the negative part of the Zariski decomposition of $-(K_{X_\eta}+\Delta_\eta)$ as a boundary after specialization; a family version of the Zariski decomposition in higher dimensions would be the natural route to extend the dimension-2 theorem.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies generic invariance of the Fano type property in families of projective pairs. Theorem 1.2 asserts equivalence between (a) existence of a Zariski-dense set of closed fibers that are Fano type with constant anti-canonical volume v>0, and (b) the geometric generic fiber is Fano type, over any algebraically closed field of characteristic 0. Theorem 1.3 proves Conjecture 1.1 for Fano type surfaces. Corollary 1.4 derives the DCC property for anti-canonical volumes of bounded families of Fano type surfaces. Theorem 1.6 gives a reduction-mod-p analogue for globally F-regular type. The proofs combine a log canonical threshold criterion for Fano type (Lemma 5.2), quasi-monomial valuation computations after Xu, and volume semicontinuity results (Theorem 4.2 from Jiao and Theorem 4.3). The reverse direction of Theorem 1.2 spreads a generic-fiber MMP to all fibers and uses Lemmas 5.3 and 5.4.

Significance. If the proofs can be completed, Theorem 1.2 would be a strong and useful criterion: pointwise Fano type plus constancy of anti-canonical volume would force the generic fiber to be Fano type, and would imply Conjecture 1.1 in this case. The lct-based strategy and the positive-characteristic parallel are natural and elegant. Theorem 1.3 and the explicit counterexample in Example 5.6 are valuable contributions. The paper is not circular: the target statements are not used as inputs, and the external theorems cited are independent of the main claims. The main caveat is that the (b)⇒(a) direction relies on an unproved fiberwise MMP assertion (Remark 5.5), and the volume-semicontinuity foundation includes an unreviewed preprint and a terse proof; the stated generality is therefore not yet established.

major comments (2)
  1. [§5, proof of Theorem 1.2, (b)⇒(a), display (5.4); see also display (5.9) and Remark 5.5] The proof asserts that after spreading a −(K_{X_η}+Δ_η)-MMP over S, for every closed fiber the restricted sequence (5.4) is a sequence of −(K_{X_{(i-1)s}}+Δ_{(i-1)s})-negative maps, and then applies Lemma 5.3 and Lemma 5.4 to conclude that each X_{is} and Y_s is Fano type. This inference is not justified: Lemma 5.3 requires actual MMP steps, and an arbitrary D-negative birational map need not preserve Fano type or anti-canonical volumes. A generic-fiber MMP step need not restrict to an MMP step on every closed fiber, because the contracted locus, the extremal ray, and the induced flip can behave differently on special fibers. Remark 5.5 explicitly concedes that the statement is only 'anticipated' and points to [9, Lemma 5.5] for S a variety over C, whereas Theorem 1.2 is stated for arbitrary algebraically closed fields of characteristic 0. The same gap affects the final volume-constancy conclusion and therefore Corollary 1.4, and it also affects the analogous display (5.9) in the proof of Theorem 1.6. This load-bearing premise needs a proof, or the theorems need to be restricted to the setting where it is known.
  2. [§4, Theorem 4.3 and the use of Theorem 4.2] The paper relies on [30, Theorem 1.1] for the volume semicontinuity that underpins Theorem 1.2, but [30] is an unreviewed arXiv preprint and its statement is not reproduced. The proof of the analog Theorem 4.3 contains a terse and partly illegible inequality chain: after combining (4.1)–(4.3) the text writes 'A^{d-1}_s·(A_s+E_s)≤(A_s+E_s)≤A^d_s+C′√ε', where the middle term is not a number and is dimensionally inconsistent; the intended intersection-theoretic inequality must be supplied. The passage to the limit as ε and ε′ tend to 0, and in Case 1 the interchange of inf over s∈S′ with the limit t→0+, are also only sketched. Since Theorem 4.3 is the basis for Theorem 1.6 and Theorem 4.2 for Theorem 1.2, these gaps need to be closed, or the external theorem should be quoted with a complete proof.
minor comments (4)
  1. [§2.4] The text contains a stray '=skip' marker between the heading 'Nakayama’s asymptotic order' and the following paragraph; this appears to be an editing artifact and should be removed.
  2. [Abstract and §1] The phrase 'reduction modp' should be typeset as 'reduction mod p'; the spacing error appears in the abstract and in the introduction.
  3. [§5, Lemma 5.2] The 'if' direction of Lemma 5.2 is dismissed as being the same as [47, Lemma 3.1], but it is used in both main theorems; a self-contained proof or a precise statement of the referenced lemma would improve the paper.
  4. [§5, proof of Theorem 1.2, (a)⇒(b)] In the chain of inequalities following display (5.2), the line 'vol(D_s)=vol(D_η)' uses the hypothesis that the volume is a constant v on S′ together with Theorem 4.2; this step should be spelled out, since Theorem 4.2 only gives vol(D_η)=inf_{s∈S′} vol(D_s) in general.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's derivations are self-contained against external results, and the one flagged weakness (Remark 5.5) is an unproved gap, not a circular reduction.

full rationale

The claimed derivation chain does not reduce to its own inputs. In the (a) implies (b) direction of Theorem 1.2, the Fano type conclusion for (X_eta, Delta_eta) is obtained by comparing log canonical thresholds through inequalities (5.1) and (5.3), which are established from Proposition 2.5 (discrepancy comparison), Lemma 2.16 (weighted blow-ups), Lemma 2.11 (volume drop vs. asymptotic order), Lemma 2.13 (semicontinuity), Theorem 2.15 (quasi-monomial valuation computing lct_sigma), and Theorem 4.2 (Jiao's volume semicontinuity). No Fano type property of the generic fiber is assumed anywhere in that chain, and no parameter is fitted to the target conclusion. In the (b) implies (a) direction, the proof uses existence of MMP [3], Lemma 5.3 and Lemma 5.4, and the constancy of volumes is derived from cohomology vanishing and the map (5.4). The only genuinely weak point is display (5.4): the paper asserts that the restricted sequence is a sequence of -(K+Delta)-negative maps and treats them as MMP steps on every closed fiber, while Remark 5.5 explicitly states that the stronger assertion that (5.4) is a -(K_{X_s}+Delta_s)-MMP is only 'anticipated' and refers to [9, Lemma 5.5] when S is a variety over C. This is an honest admission of an unproved premise and a correctness gap in the stated generality, not circularity: the assertion is not an input to the proof, it is not justified by the paper's own conclusion, and the cited support is an external result by different authors rather than a self-citation. The same pattern holds for Theorem 1.6, where Theorem 4.3, [24], [40], and [41] provide independent ingredients. There are no cases of fitted input called prediction, no self-citation load-bearing steps, no uniqueness theorem imported from the authors' own prior work, and no ansatz smuggled in via citation. The paper even flags its own limitation in Remark 5.5, which supports the conclusion that the derivation is not circular but incomplete in the stated generality.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim depends on standard MMP results, a recent volume semicontinuity theorem of Jiao quoted from [30], and an unproved fiberwise MMP bridge flagged in Remark 5.5. No free parameters or invented entities are used.

assumptions (4)
  • standard math Standard MMP results: existence of D-MMP for Fano type pairs, klt contraction theorem, Q-factorialization, Kodaira vanishing.
    Invoked in Lemma 5.3, Lemma 5.4, and the proofs of Theorems 1.2 and 1.6.
  • domain assumption Jiao's theorem [30, Theorem 1.1]: for a flat family over characteristic 0, vol(D_eta)=inf_{s in S'} vol(D_s).
    Quoted in Theorem 4.2 and used in the (a) to (b) direction of Theorem 1.2 to infer that the anti-canonical divisor of the generic fiber is big.
  • ad hoc to paper The extended fiberwise maps (5.4) form a D-MMP and preserve Fano type and anti-canonical volumes.
    Remark 5.5 states this is anticipated and cites [9, Lemma 5.5] only for S over C, while the main theorem is stated over arbitrary algebraically closed characteristic zero fields.
  • standard math Xu's quasi-monomial valuation theorem [46, Theorem 1.1] and the lct-sigma criterion [47, Lemma 3.1].
    Used to reduce log canonical threshold computations to a single log resolution and to characterize Fano type pairs.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On volumes and the generic invariance of Fano type varieties." pith.science (2026). https://pith.science/paper/R674KZ52

@misc{pith2026250613603,
  author       = {Pith},
  title        = {Pith review of: On volumes and the generic invariance of Fano type varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R674KZ52}},
  note         = {Machine review of arXiv:2506.13603}
}
abstract

We demonstrate the generic invariance of the Fano type property in cases where the volumes of anti-canonical divisors of Fano type fibers are a constant over a Zariski-dense subset, or the Fano type fibers are dimension $2$. Additionally, paralleling this theorem, we establish a conjecture by Schwede and Smith under the condition that the volumes of anti-canonical divisors remain constant in the reduction mod $p$.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [30]

    Junpeng Jiao,The volume function is upper semicontinuous on families of divisors, arXiv preprint arXiv:2504.16676 (2025)

    work page Pith review arXiv 2025

  2. [1]

    Caucher Birkar,Anti-pluricanonical systems on Fano varieties, Ann. of Math. (2)190(2019), no. 2, 345–463. MR3997127

    work page 2019

  3. [2]

    ,Boundedness of Fano type fibrations, Ann. Sci. ´Ec. Norm. Sup´ er. (4)57(2024), no. 3, 787–840. MR4773297

    work page 2024

  4. [3]

    Hacon, and James McKernan,Existence of minimal models for varieties of log general type, J

    Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan,Existence of minimal models for varieties of log general type, J. Amer. Math. Soc.23(2010), no. 2, 405–468. MR2601039 ON VOLUMES AND THE GENERIC INV ARIANCE OF F ANO TYPE V ARIETIES 27

    work page 2010

  5. [4]

    J.171(2022), no

    Harold Blum, Yuchen Liu, and Chenyang Xu,Openness of K-semistability for Fano varieties, Duke Math. J.171(2022), no. 13, 2753–2797. MR4505846

    work page 2022

  6. [5]

    Ann.347(2010), no

    Manuel Blickle, Karl Schwede, Shunsuke Takagi, and Wenliang Zhang,Discreteness and rationality ofF-jumping numbers on singular varieties, Math. Ann.347(2010), no. 4, 917–949. MR2658149

    work page 2010

  7. [6]

    Algebraic Geom

    S´ ebastien Boucksom, Jean-Pierre Demailly, Mihai P˘ aun, and Thomas Peternell,The pseudo-effective cone of a compact K¨ ahler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom. 22(2013), no. 2, 201–248. MR3019449

    work page 2013

  8. [7]

    Eric Canton,Berkovich log discrepancies in positive characteristic, Pure Appl. Math. Q.16(2020), no. 5, 1465–1532. MR4221003

    work page 2020

Show all 48 references
  1. [8]

    Paolo Cascini, Tatsuro Kawakami, and Shunsuke Takagi,Threefolds of globally F-regular type with nef anti-canonical divisor, arXiv preprint arXiv:2410.03871 (2024)

    2024 arXiv

  2. [9]

    Sung Rak Choi, Zhan Li, and Chuyu Zhou,Variation of cones of divisors in a family of varieties–Fano type case, arXiv preprint arXiv:2504.04109 (2025)

    2025 arXiv

  3. [10]

    Tommaso de Fernex, Lawrence Ein, and Mircea Musta¸t˘ a,Log canonical thresholds on varieties with bounded singularities, Classification of algebraic varieties, 2011, pp. 221–257. MR2779474

    2011

  4. [11]

    A. J. de Jong et al.,The stacks project

  5. [12]

    Olivier Debarre,Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York,

  6. [13]

    Japan Acad

    Osamu Fujino,Semi-stable minimal model program for varieties with trivial canonical divisor, Proc. Japan Acad. Ser. A Math. Sci.87(2011), no. 3, 25–30. MR2802603

    2011

  7. [14]

    Japan Acad

    Takao Fujita,On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci.55(1979), no. 3, 106–110. MR531454

    1979

  8. [15]

    J.17(1994), no

    ,Approximating Zariski decomposition of big line bundles, Kodai Math. J.17(1994), no. 1, 1–3. MR1262949

    1994

  9. [16]

    J.65(2016), no

    Mihai Fulger, J´ anos Koll´ ar, and Brian Lehmann,Volume and Hilbert function ofR-divisors, Michigan Math. J.65(2016), no. 2, 371–387. MR3510912

    2016

  10. [17]

    Math.280(2015), 47–78

    Yoshinori Gongyo, Zhiyuan Li, Zsolt Patakfalvi, Karl Schwede, Hiromu Tanaka, and Runhong Zong, On rational connectedness of globally F -regular threefolds, Adv. Math.280(2015), 47–78. MR3350212

    2015

  11. [18]

    Algebraic Geom.24(2015), no

    Yoshinori Gongyo, Shinnosuke Okawa, Akiyoshi Sannai, and Shunsuke Takagi,Characterization of varieties of Fano type via singularities of Cox rings, J. Algebraic Geom.24(2015), no. 1, 159–182. MR3275656

    2015

  12. [19]

    Yoshinori Gongyo and Shunsuke Takagi,Surfaces of globally F -regular and F -split type, Math. Ann. 364(2016), no. 3-4, 841–855. MR3466854

    2016

  13. [20]

    Grothendieck, ´El´ ements de G´ eom´ etrie Alg´ ebrique

    A. Grothendieck, ´El´ ements de G´ eom´ etrie Alg´ ebrique. IV. ´ etude locale des sch´ emas et des morphismes de sch´ emas. II, Inst. Hautes´Etudes Sci. Publ. Math.24(1965), 231. MR199181

    1965

  14. [21]

    Hacon, James McKernan, and Chenyang Xu,On the birational automorphisms of varieties of general type, Ann

    Christopher D. Hacon, James McKernan, and Chenyang Xu,On the birational automorphisms of varieties of general type, Ann. of Math. (2)177(2013), no. 3, 1077–1111

    2013

  15. [22]

    Hacon, James McKernan, and Chenyang Xu,ACC for log canonical thresholds, Ann

    Christopher D. Hacon, James McKernan, and Chenyang Xu,ACC for log canonical thresholds, Ann. of Math. (2)180(2014), no. 2, 523–571. MR3224718

    2014

  16. [23]

    ,Boundedness of moduli of varieties of general type, J. Eur. Math. Soc. (JEMS)20(2018), no. 4, 865–901. MR3779687

    2018

  17. [24]

    log terminal and log canonical singularities, J

    Nobuo Hara and Kei-Ichi Watanabe,F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom.11(2002), no. 2, 363–392. MR1874118

    2002

  18. [25]

    Robin Hartshorne,Algebraic geometry, Graduate Texts in Mathematics, vol. No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR463157

    1977

  19. [26]

    Melvin Hochster and Craig Huneke,Tight closure and strong F -regularity, 1989, pp. 119–133. Colloque en l’honneur de Pierre Samuel (Orsay, 1987). MR1044348

    1989

  20. [27]

    ,Tight closure in equal characteristic zero, preprint (1999). 28 D. KIM

    1999

  21. [28]

    336, Cambridge University Press, Cambridge, 2006

    Craig Huneke and Irena Swanson,Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR2266432

    2006

  22. [29]

    Z.280(2015), no

    DongSeon Hwang and Jinhyung Park,Characterization of log del Pezzo pairs via anticanonical models, Math. Z.280(2015), no. 1-2, 211–229. MR3343904

    2015

  23. [31]

    Mattias Jonsson and Mircea Musta t,˘ a,Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble)62(2012), no. 6, 2145–2209. MR3060755

    2012

  24. [32]

    231, Cambridge University Press, Cambridge, 2023

    J´ anos Koll´ ar,Families of varieties of general type, Cambridge Tracts in Mathematics, vol. 231, Cambridge University Press, Cambridge, 2023. With the collaboration of Klaus Altmann and S´ andor J. Kov´ acs. MR4566297

    2023

  25. [33]

    134, Cambridge University Press, Cambridge, 1998

    J´ anos Koll´ ar and Shigefumi Mori,Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. MR1658959

    1998

  26. [34]

    I, Ergebnisse der Mathematik und ihrer Gren- zgebiete

    Robert Lazarsfeld,Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Gren- zgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-V...

    2004

  27. [35]

    II, Ergebnisse der Mathematik und ihrer Grenzgebiete

    ,Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 200...

    2004

  28. [36]

    Chi Li and Chenyang Xu,Stability of valuations and Koll´ ar components, J. Eur. Math. Soc. (JEMS) 22(2020), no. 8, 2573–2627. MR4118616

    2020

  29. [37]

    14, Mathematical Society of Japan, Tokyo, 2004

    Noboru Nakayama,Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR2104208

    2004

  30. [38]

    Shinnosuke Okawa,Surfaces of globally F -regular type are of Fano type, Tohoku Math. J. (2)69 (2017), no. 1, 35–42. MR3640012

    2017

  31. [39]

    Z.265(2010), no

    Karl Schwede,Centers ofF-purity, Math. Z.265(2010), no. 3, 687–714. MR2644316

    2010

  32. [40]

    Smith,Globally F -regular and log Fano varieties, Adv

    Karl Schwede and Karen E. Smith,Globally F -regular and log Fano varieties, Adv. Math.224 (2010), no. 3, 863–894. MR2628797

    2010

  33. [41]

    Smith,Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, 2000, pp

    Karen E. Smith,Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, 2000, pp. 553–572. Dedicated to William Fulton on the occasion of his 60th birthday. MR1786505

    2000

  34. [42]

    Ann.378(2020), no

    Xiaotao Sun and Mingshuo Zhou,Globally F -regular type of moduli spaces, Math. Ann.378(2020), no. 3-4, 1245–1270. MR4163526

    2020

  35. [43]

    Satoshi Takagi,Fujita’s approximation theorem in positive characteristics, J. Math. Kyoto Univ.47 (2007), no. 1, 179–202. MR2359108

    2007

  36. [44]

    Algebraic Geom.13 (2004), no

    Shunsuke Takagi,An interpretation of multiplier ideals via tight closure, J. Algebraic Geom.13 (2004), no. 2, 393–415. MR2047704

    2004

  37. [45]

    Sigma12(2024), Paper No

    Jianping Wang and Xueqing Wen,Globally F-regular type of the moduli spaces of parabolic symplec- tic/orthogonal bundles on curves, Forum Math. Sigma12(2024), Paper No. e69, 21. MR4749934

    2024

  38. [46]

    Chenyang Xu,A minimizing valuation is quasi-monomial, Ann. of Math. (2)191(2020), no. 3, 1003–1030. MR4088355

    2020

  39. [47]

    Alg´ ebrique (2023), Art

    ,K-stability for varieties with a big anticanonical class, ´Epijournal G´ eom. Alg´ ebrique (2023), Art. 7, 9. MR4671736

    2023

  40. [48]

    50, Cambridge University Press, Cambridge, 2025

    ,K-stability of Fano varieties, New Mathematical Monographs, vol. 50, Cambridge University Press, Cambridge, 2025. MR4893062 ON VOLUMES AND THE GENERIC INV ARIANCE OF F ANO TYPE V ARIETIES 29 (Donghyeon Kim)Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seodaemun-...

    2025

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.