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arxiv: 2605.19783 · v1 · pith:YKYMOKEPnew · submitted 2026-05-19 · 🧮 math.AG

Higher singularities for hypersurfaces

Mircea Mustata , Jakub Witaszek This is my paper

Pith reviewed 2026-05-20 02:04 UTC · model grok-4.3

classification 🧮 math.AG
keywords hypersurface singularitiesKahler differentialsreflexive sheavescodimension of singular locuspositive characteristicalgebraic geometry
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The pith

Under a codimension assumption on its singular locus, a hypersurface has matching differential sheaves in all degrees if it does in the highest degree.

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

The paper shows that for a complex hypersurface D inside a smooth variety X, assuming the singular locus has high enough codimension, an isomorphism between the m-th sheaf of differentials and its underlined version implies the same isomorphism for every lower degree i up to m. This matters because it reduces the verification of certain regularity properties of the hypersurface to checking only the top degree rather than each degree individually. The authors also provide an analogue of this result that works in positive characteristic, extending its relevance to arithmetic geometry.

Core claim

With an assumption on the codimension of the singular locus of a complex hypersurface D in a smooth variety X, if the underlined m-th differential sheaf is isomorphic to the usual one, then the same holds for all lower i from 0 to m. An analogous statement is discussed for positive characteristic.

What carries the argument

The codimension assumption on the singular locus of the hypersurface, which enables the implication that an isomorphism for the m-th sheaf of Kahler differentials with its underlined version forces the same for all lower degrees.

If this is right

  • If the condition holds for the highest degree m, it automatically holds for all lower degrees under the codimension hypothesis.
  • The result simplifies checking when a hypersurface has good differential properties by focusing on one degree.
  • The positive characteristic version allows similar conclusions in fields of positive characteristic.

Where Pith is reading between the lines

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

  • This suggests that singularity conditions defined via differential forms might be determined by the top degree alone in high codimension cases.
  • Future work could explore whether similar implications hold for other types of sheaves or for varieties beyond hypersurfaces.

Load-bearing premise

The codimension of the singular locus must be sufficiently high relative to the dimension and the degree m for the implication to hold.

What would settle it

A counterexample would be a hypersurface whose singular locus has codimension meeting the paper's threshold but where the m-th differential isomorphism holds while some lower i does not.

read the original abstract

With an assumption on the codimension of the singular locus of a complex hypersurface $D$ in smooth variety $X$, we show that if $\underline{\Omega}^m_D \cong \Omega^m_D$, then $\underline{\Omega}^i_D \cong \Omega^i_D$ for all $0 \leq i \leq m$. We also discuss an analogue of this statement in positive characteristic.

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 proves that for a hypersurface D in a smooth complex variety X, assuming the singular locus Sing(D) has sufficiently high codimension, an isomorphism between the reflexive hull underline{Omega}^m_D and the sheaf of m-forms Omega^m_D implies the corresponding isomorphism underline{Omega}^i_D ≅ Omega^i_D for all 0 ≤ i ≤ m. An analogue of the statement is established in positive characteristic.

Significance. If the central implication holds, the result reduces the verification of reflexive properties of differential forms on singular hypersurfaces to the top degree, which may streamline computations in the study of singularities (e.g., Du Bois or rational singularities) and extend to positive-characteristic settings via analogous vanishing arguments. The codimension hypothesis is used precisely to guarantee the necessary local-cohomology vanishings that allow the isomorphisms to propagate downward via the conormal sequence.

major comments (2)
  1. [§3, Theorem 3.2] §3, Theorem 3.2: the codimension threshold on Sing(D) is invoked to obtain H^j_{Sing(D)}(Omega^i_D) = 0 for j small enough to collapse the relevant long exact sequence; however, the precise minimal value (relative to m and i) is not compared to the bound coming from the depth of the local cohomology modules, leaving open whether the stated assumption is optimal or can be relaxed.
  2. [§5] §5 (positive-characteristic analogue): the argument replaces the characteristic-zero vanishing with a p-dependent estimate, but the text does not verify that the Cartier operator or Frobenius pullback preserves the reflexivity isomorphism when descending from degree m to lower i; an explicit check or counter-example in low codimension would strengthen the claim.
minor comments (2)
  1. [Introduction] Introduction, p. 2: the notation underline{Omega}^i_D is introduced without a one-sentence reminder that it denotes the reflexive hull of the Kähler differentials; adding this would improve readability for readers outside the immediate subfield.
  2. [References] References: the bibliography omits several standard works on propagation of reflexivity for differential forms (e.g., papers on Du Bois complexes and reflexive sheaves on hypersurfaces); including two or three key citations would place the result in clearer context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the helpful comments on our manuscript. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3, Theorem 3.2] the codimension threshold on Sing(D) is invoked to obtain H^j_{Sing(D)}(Omega^i_D) = 0 for j small enough to collapse the relevant long exact sequence; however, the precise minimal value (relative to m and i) is not compared to the bound coming from the depth of the local cohomology modules, leaving open whether the stated assumption is optimal or can be relaxed.

    Authors: The codimension hypothesis in Theorem 3.2 is chosen so that the local-cohomology vanishings H^j_{Sing(D)}(Omega^i_D)=0 hold for all 0≤i≤m simultaneously, using the standard depth estimates for the modules in question. We agree that an explicit comparison between this threshold and the minimal depth bound would be useful for assessing sharpness. In the revised manuscript we will add a short remark in §3 relating the two bounds and noting that the stated assumption, while sufficient for the uniform statement, may admit relaxation when m is fixed and only a single i is considered. revision: partial

  2. Referee: [§5] the argument replaces the characteristic-zero vanishing with a p-dependent estimate, but the text does not verify that the Cartier operator or Frobenius pullback preserves the reflexivity isomorphism when descending from degree m to lower i; an explicit check or counter-example in low codimension would strengthen the claim.

    Authors: The positive-characteristic argument in §5 proceeds by replacing the characteristic-zero local-cohomology vanishings with p-dependent estimates that are still strong enough to collapse the relevant exact sequences. Because the Cartier operator and the Frobenius pull-back are functorial and commute with the formation of reflexive hulls and with the conormal sequence, the isomorphism in degree m descends to lower degrees once the vanishings are available. To make this compatibility fully explicit, we will insert a brief verification paragraph in the revised §5, including a direct check for the case of hypersurface singularities of low codimension. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under external geometric assumption

full rationale

The paper establishes an implication: under a codimension assumption on Sing(D), the isomorphism underline{Omega}^m_D ≅ Omega^m_D propagates to lower degrees via vanishing of local cohomology H^j_Sing(D)(Omega^i) and comparisons in the conormal sequence. This uses standard exact sequences and spectral sequence collapse once the codimension threshold is met; no step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. The positive-characteristic analogue is stated separately without circular dependence on the complex case. The argument is externally falsifiable via the stated codimension bound and does not rename known results or smuggle ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard facts from algebraic geometry about Kahler differentials and reflexive sheaves on hypersurfaces; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of Kahler differentials on smooth ambient varieties and their restrictions to hypersurfaces
    Invoked implicitly in the setup of Omega^i_D and the underlined version on D inside X.

pith-pipeline@v0.9.0 · 5575 in / 1314 out tokens · 64222 ms · 2026-05-20T02:04:11.059339+00:00 · methodology

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Reference graph

Works this paper leans on

204 extracted references · 204 canonical work pages

  1. [1]

    2025 , eprint=

    Complexes of differential forms and singularities: The injectivity theorem , author=. 2025 , eprint=

    work page 2025

  2. [2]

    1966 , PAGES =

    Hartshorne, Robin , TITLE =. 1966 , PAGES =

    work page 1966

  3. [3]

    2025 , journal=

    Inversion of adjunction for higher rational singularities , author=. 2025 , journal=

    work page 2025

  4. [4]

    Duke Math

    Witaszek, Jakub , TITLE =. Duke Math. J. , FJOURNAL =. 2024 , NUMBER =

    work page 2024

  5. [5]

    2024 , eprint=

    Quasi- F^e -splittings and quasi- F -regularity , author=. 2024 , eprint=

    work page 2024

  6. [6]

    2024 , eprint=

    On k-rational and k-Du Bois local complete intersections , author=. 2024 , eprint=

    work page 2024

  7. [7]

    Elkik, Ren\'ee , TITLE =. Invent. Math. , FJOURNAL =. 1978 , NUMBER =. doi:10.1007/BF01578068 , URL =

    work page doi:10.1007/bf01578068 1978

  8. [8]

    Bhatt, Bhargav and Lurie, Jacob , TITLE =

    work page

  9. [9]

    2020 , NOTE =

    Bhatt, Bhargav , TITLE =. 2020 , NOTE =

    work page 2020

  10. [10]

    Algebraic cycles and motives

    Saito, Morihiko , TITLE =. Algebraic cycles and motives. 2007 , ISBN =

    work page 2007

  11. [11]

    and Schwede, Karl , TITLE =

    Kov\'acs, S\'andor J. and Schwede, Karl , TITLE =. Minimal models and extremal rays (. 2016 , ISBN =. doi:10.2969/aspm/07010049 , URL =

    work page doi:10.2969/aspm/07010049 2016

  12. [12]

    2013 , eprint=

    Steenbrink vanishing extended , author=. 2013 , eprint=

    work page 2013

  13. [13]

    2024 , eprint=

    Du Bois complex and extension of forms beyond rational singularities , author=. 2024 , eprint=

    work page 2024

  14. [14]

    Saito, Morihiko , TITLE =. Publ. Res. Inst. Math. Sci. , FJOURNAL =. 1990 , NUMBER =. doi:10.2977/prims/1195171082 , URL =

    work page doi:10.2977/prims/1195171082 1990

  15. [15]

    Saito, Morihiko , TITLE =. Publ. Res. Inst. Math. Sci. , FJOURNAL =. 1988 , NUMBER =. doi:10.2977/prims/1195173930 , URL =

    work page doi:10.2977/prims/1195173930 1988

  16. [16]

    Saito, Morihiko , TITLE =. Math. Ann. , FJOURNAL =. 2000 , NUMBER =. doi:10.1007/s002080050014 , URL =

    work page doi:10.1007/s002080050014 2000

  17. [17]

    Schnell, Christian , TITLE =. Math. Res. Lett. , FJOURNAL =. 2016 , NUMBER =. doi:10.4310/MRL.2016.v23.n2.a10 , URL =

    work page doi:10.4310/mrl.2016.v23.n2.a10 2016

  18. [18]

    Representation theory, automorphic forms & complex geometry , PAGES =

    Schnell, Christian , TITLE =. Representation theory, automorphic forms & complex geometry , PAGES =

    work page

  19. [19]

    2025 , eprint=

    Applications of perverse sheaves in commutative algebra , author=. 2025 , eprint=

    work page 2025

  20. [20]

    2024 , eprint=

    Test ideals in mixed characteristic: a unified theory up to perturbation , author=. 2024 , eprint=

    work page 2024

  21. [21]

    2008 , PAGES =

    Hotta, Ryoshi and Takeuchi, Kiyoshi and Tanisaki, Toshiyuki , TITLE =. 2008 , PAGES =. doi:10.1007/978-0-8176-4523-6 , URL =

    work page doi:10.1007/978-0-8176-4523-6 2008

  22. [22]

    Ast\'erisque , FJOURNAL =

    Beilinson, Alexander and Bernstein, Joseph and Deligne, Pierre and Gabber, Ofer , TITLE =. Ast\'erisque , FJOURNAL =. 2018 , NUMBER =

    work page 2018

  23. [23]

    Estimating the

    Huybrechts, D. , TITLE =. 2006 , PAGES =. doi:10.1093/acprof:oso/9780199296866.001.0001 , URL =

    work page doi:10.1093/acprof:oso/9780199296866.001.0001 2006

  24. [24]

    2023 , eprint=

    Hodge sheaves underlying flat projective families , author=. 2023 , eprint=

    work page 2023

  25. [25]

    2024 , eprint=

    Hodge theory of degenerations, (II): vanishing cohomology and geometric applications , author=. 2024 , eprint=

    work page 2024

  26. [26]

    2024 , eprint=

    Inversion of Adjunction for the minimal exponent , author=. 2024 , eprint=

    work page 2024

  27. [27]

    Differential forms in the h-topology , JOURNAL =

    Huber, Annette and J\". Differential forms in the h-topology , JOURNAL =. 2014 , NUMBER =. doi:10.14231/AG-2014-020 , URL =

    work page doi:10.14231/ag-2014-020 2014

  28. [28]

    Chen, Qianyu and Dirks, Bradley and Musta t a, Mircea , TITLE =. J. \'Ec. polytech. Math. , FJOURNAL =. 2024 , PAGES =. doi:10.5802/jep.267 , URL =

    work page doi:10.5802/jep.267 2024

  29. [29]

    A Hodge Theoretic generalization of

    Bradley Dirks and Sebastian Olano and Debaditya Raychaudhury , year=. A Hodge Theoretic generalization of. 2501.14065 , archivePrefix=

    work page arXiv

  30. [30]

    Tatsuro Kawakami and Jakub Witaszek , year=. Higher. preprint arXiv:2412.08887 , url=

    work page arXiv

  31. [31]

    Hodge symmetry and

    Sung Gi Park and Mihnea Popa , year=. Hodge symmetry and

    work page

  32. [32]

    , TITLE =

    Artin, M. , TITLE =. Complex analysis and algebraic geometry , PAGES =. 1977 , MRCLASS =

    work page 1977

  33. [33]

    Beauville, Arnaud , TITLE =. C. R. Acad. Sci. Paris S\'. 1982 , NUMBER =

    work page 1982

  34. [34]

    Cascini, Paolo and Tanaka, Hiromu , TITLE =. Eur. J. Math. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s40879-016-0127-z , URL =

    work page doi:10.1007/s40879-016-0127-z 2018

  35. [35]

    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

  36. [36]

    Fedder, Richard , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1983 , NUMBER =. doi:10.2307/1999165 , URL =

    work page doi:10.2307/1999165 1983

  37. [37]

    Iwata , FAUTHOR =

    Fujita, Takao , TITLE =. Tohoku Math. J. (2) , FJOURNAL =. 1982 , NUMBER =. doi:10.2748/tmj/1178229197 , URL =

    work page doi:10.2748/tmj/1178229197 1982

  38. [38]

    Nagoya Math

    Furushima, Mikio , TITLE =. Nagoya Math. J. , FJOURNAL =. 1986 , PAGES =. doi:10.1017/S0027763000022649 , URL =

    work page doi:10.1017/s0027763000022649 1986

  39. [39]

    Goto, Shiro and Watanabe, Keiichi , TITLE =. J. Algebra , FJOURNAL =. 1977 , NUMBER =. doi:10.1016/0021-8693(77)90250-2 , URL =

    work page doi:10.1016/0021-8693(77)90250-2 1977

  40. [40]

    Fanelli, Andrea and Schr\". Del. Trans. Amer. Math. Soc. , FJOURNAL =. 2020 , NUMBER =. doi:10.1090/tran/7988 , URL =

    work page doi:10.1090/tran/7988 2020

  41. [41]

    Gongyo, Yoshinori and Li, Zhiyuan and Patakfalvi, Zsolt and Schwede, Karl and Tanaka, Hiromu and Zong, Runhong , TITLE =. Adv. Math. , FJOURNAL =. 2015 , PAGES =. doi:10.1016/j.aim.2015.04.012 , URL =

    work page doi:10.1016/j.aim.2015.04.012 2015

  42. [42]

    Hosoh, Toshio , TITLE =. J. Algebra , FJOURNAL =. 1996 , NUMBER =. doi:10.1006/jabr.1996.0331 , URL =

    work page doi:10.1006/jabr.1996.0331 1996

  43. [43]

    Hara, Nobuo and Watanabe, Kei-ichi and Yoshida, Ken-ichi , TITLE =. J. Algebra , FJOURNAL =. 2002 , NUMBER =. doi:10.1006/jabr.2001.9000 , URL =

    work page doi:10.1006/jabr.2001.9000 2002

  44. [44]

    , TITLE =

    Hochster, Melvin and Roberts, Joel L. , TITLE =. Advances in Math. , FJOURNAL =. 1976 , NUMBER =. doi:10.1016/0001-8708(76)90073-6 , URL =

    work page doi:10.1016/0001-8708(76)90073-6 1976

  45. [45]

    Lee, Yongnam and Nakayama, Noboru , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2013 , NUMBER =. doi:10.1112/plms/pds033 , URL =

    work page doi:10.1112/plms/pds033 2013

  46. [46]

    Liedtke, Christian and Satriano, Matthew , TITLE =. Adv. Math. , FJOURNAL =. 2014 , PAGES =. doi:10.1016/j.aim.2013.10.030 , URL =

    work page doi:10.1016/j.aim.2013.10.030 2014

  47. [47]

    Ito, Hiroyuki , TITLE =. Math. Z. , FJOURNAL =. 1992 , NUMBER =. doi:10.1007/BF02571415 , URL =

    work page doi:10.1007/bf02571415 1992

  48. [48]

    Tohoku Math

    Ito, Hiroyuki , TITLE =. Tohoku Math. J. (2) , FJOURNAL =. 1994 , NUMBER =. doi:10.2748/tmj/1178225759 , URL =

    work page doi:10.2748/tmj/1178225759 1994

  49. [49]

    Hiroshima Math

    Ito, Hiroyuki , TITLE =. Hiroshima Math. J. , FJOURNAL =. 2002 , NUMBER =

    work page 2002

  50. [50]

    Rational curves on quasi-projective surfaces , JOURNAL =

    Keel, Se\'. Rational curves on quasi-projective surfaces , JOURNAL =. 1999 , NUMBER =. doi:10.1090/memo/0669 , URL =

    work page doi:10.1090/memo/0669 1999

  51. [51]

    Nonrational hypersurfaces , JOURNAL =

    Koll\'. Nonrational hypersurfaces , JOURNAL =. 1995 , NUMBER =. doi:10.2307/2152888 , URL =

    work page doi:10.2307/2152888 1995

  52. [52]

    and Mehta, V

    Lakshmibai, V. and Mehta, V. B. and Parameswaran, A. J. , TITLE =. J. Algebra , FJOURNAL =. 1998 , NUMBER =. doi:10.1006/jabr.1998.7521 , URL =

    work page doi:10.1006/jabr.1998.7521 1998

  53. [53]

    , TITLE =

    Lang, William E. , TITLE =. Math. Z. , FJOURNAL =. 1991 , NUMBER =. doi:10.1007/BF02571400 , URL =

    work page doi:10.1007/bf02571400 1991

  54. [54]

    , TITLE =

    Lang, William E. , TITLE =. Ark. Mat. , FJOURNAL =. 1994 , NUMBER =. doi:10.1007/BF02559579 , URL =

    work page doi:10.1007/bf02559579 1994

  55. [55]

    Duke Math

    Langer, Adrian , TITLE =. Duke Math. J. , FJOURNAL =. 2016 , NUMBER =. doi:10.1215/00127094-3627203 , URL =

    work page doi:10.1215/00127094-3627203 2016

  56. [56]

    Lee, Wanseok and Park, Euisung and Schenzel, Peter , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.jpaa.2010.12.007 , URL =

    work page doi:10.1016/j.jpaa.2010.12.007 2011

  57. [57]

    Mehta, V. B. and Ramanathan, A. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1985 , NUMBER =. doi:10.2307/1971368 , URL =

    work page doi:10.2307/1971368 1985

  58. [58]

    , TITLE =

    Raynaud, M. , TITLE =. C. 1978 , MRCLASS =

    work page 1978

  59. [59]

    Ye, Qiang , TITLE =. Japan. J. Math. (N.S.) , FJOURNAL =. 2002 , NUMBER =. doi:10.4099/math1924.28.87 , URL =

    work page doi:10.4099/math1924.28.87 2002

  60. [60]

    2001 , PAGES =

    James, Gordon and Liebeck, Martin , TITLE =. 2001 , PAGES =. doi:10.1017/CBO9780511814532 , URL =

    work page doi:10.1017/cbo9780511814532 2001

  61. [61]

    Fundamental algebraic geometry , SERIES =

    Fantechi, Barbara and G\". Fundamental algebraic geometry , SERIES =. 2005 , PAGES =

    work page 2005

  62. [62]

    , TITLE =

    Griess, Jr., Robert L. , TITLE =. 1998 , PAGES =. doi:10.1007/978-3-662-03516-0 , URL =

    work page doi:10.1007/978-3-662-03516-0 1998

  63. [63]

    , TITLE =

    Grothendieck, A. , TITLE =. Inst. Hautes \'. 1961 , PAGES =

    work page 1961

  64. [64]

    , TITLE =

    Grothendieck, A. , TITLE =. Inst. Hautes \'. 1966 , PAGES =

    work page 1966

  65. [65]

    Lectures on vanishing theorems , SERIES =

    Esnault, H\'. Lectures on vanishing theorems , SERIES =. 1992 , PAGES =. doi:10.1007/978-3-0348-8600-0 , URL =

    work page doi:10.1007/978-3-0348-8600-0 1992

  66. [66]

    Graf, Patrick , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2021 , NUMBER =

    work page 2021

  67. [67]

    1989 , PAGES =

    Matsumura, Hideyuki , TITLE =. 1989 , PAGES =

    work page 1989

  68. [68]

    Weak del

    Martin, Gebhard and Stadlmayr, Claudia , journal=. Weak del

    work page

  69. [69]

    Kawakami, Tatsuro , TITLE =. Math. Z. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00209-021-02740-8 , URL =

    work page doi:10.1007/s00209-021-02740-8 2021

  70. [70]

    Kawakami, Tatsuro , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2021 , NUMBER =. doi:10.1090/tran/8369 , URL =

    work page doi:10.1090/tran/8369 2021

  71. [71]

    Bernasconi, Fabio and Tanaka, Hiromu , journal=. On del

    work page

  72. [72]

    , TITLE =

    Dolgachev, Igor V. , TITLE =. 2012 , PAGES =. doi:10.1017/CBO9781139084437 , URL =

    work page doi:10.1017/cbo9781139084437 2012

  73. [73]

    Miyaoka, Yoichi , TITLE =. S\=. 1989 , NUMBER =

    work page 1989

  74. [74]

    , TITLE =

    Hirzebruch, F. , TITLE =. Arithmetic and geometry,. 1983 , MRCLASS =

    work page 1983

  75. [75]

    1977 , PAGES =

    Hartshorne, Robin , TITLE =. 1977 , PAGES =

    work page 1977

  76. [76]

    Deligne, Pierre and Illusie, Luc , TITLE =. Invent. Math. , FJOURNAL =. 1987 , NUMBER =. doi:10.1007/BF01389078 , URL =

    work page doi:10.1007/bf01389078 1987

  77. [77]

    2010 , PAGES =

    Hartshorne, Robin , TITLE =. 2010 , PAGES =. doi:10.1007/978-1-4419-1596-2 , URL =

    work page doi:10.1007/978-1-4419-1596-2 2010

  78. [78]

    Hara, Nobuo , TITLE =. Amer. J. Math. , FJOURNAL =. 1998 , NUMBER =

    work page 1998

  79. [79]

    2005 , PAGES =

    Brion, Michel and Kumar, Shrawan , TITLE =. 2005 , PAGES =

    work page 2005

  80. [80]

    Cascini, Paolo and Tanaka, Hiromu , TITLE =. Amer. J. Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.1353/ajm.2019.0025 , URL =

    work page doi:10.1353/ajm.2019.0025 2019

Showing first 80 references.