Transcendental versions in C n of the Nagata conjecture

Stephanie Nivoche

arxiv: 1906.08518 · v1 · pith:RV5SBDLVnew · submitted 2019-06-20 · 🧮 math.CV · math.AG

Transcendental versions in C n of the Nagata conjecture

Stephanie Nivoche This is my paper

Pith reviewed 2026-05-25 19:05 UTC · model grok-4.3

classification 🧮 math.CV math.AG

keywords Nagata conjecturepluripotential theorytranscendental versionsC^nmultiplicitiesplane curvescomplex analysis

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The pith

Transcendental versions of the Nagata conjecture from pluripotential theory are equivalent to a version in C^n.

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

The paper formulates new transcendental versions of the Nagata conjecture using pluripotential theory. These versions are mathematically equivalent to a formulation of the conjecture in complex n-dimensional space. This equivalence allows the algebraic problem about curve degrees and point multiplicities to be studied through analytic methods. A reader might care because it opens the conjecture to techniques from several complex variables and potential theory. The original conjecture predicts a lower bound on the degree d exceeding the square root of the number of points times the multiplicity.

Core claim

The paper formulates new transcendental versions of this conjecture coming from pluripotential theory and which are equivalent to a version in C n of the Nagata Conjecture.

What carries the argument

Transcendental versions derived from pluripotential theory that encode the multiplicity conditions analytically in C^n.

Load-bearing premise

The formulations derived from pluripotential theory are mathematically equivalent to a version of the Nagata conjecture in C^n.

What would settle it

A specific set of points and multiplicities in C^n where one version of the bound holds but the other fails.

read the original abstract

The Nagata Conjecture is one of the most intriguing open problems in the area of curves in the plane. It is easily stated. Namely, it predicts that the smallest degree d of a plane curve passing through r $\ge$ 10 general points in the projective plane P 2 with multiplicities at least l at every point, satisfies the inequality d > $\sqrt$ r $\times$ l. This conjecture has been proven by M. Nagata in 1959, if r is a perfect square greater than 9. Up to now, it remains open for every non-square r $\ge$ 10, after more than a half century of attention by many researchers. In this paper, we formulate new transcendental versions of this conjecture coming from pluripotential theory and which are equivalent to a version in C n of the Nagata Conjecture.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit pluripotential reformulations of a C^n Nagata version and proves equivalence both ways through approximation.

read the letter

The main point is that the author writes down several statements in pluripotential theory and shows they are equivalent to a version of the Nagata conjecture in C^n. The equivalence goes in both directions by approximating algebraic data with plurisubharmonic functions and the other way around, and the steps are laid out without circularity under the given hypotheses. That is the actual new content here: the concrete transcendental statements and the matching proofs. The work stays within standard pluripotential tools and does not claim to resolve the conjecture or produce new bounds. The C^n extension itself is treated as a natural move from the classical P^2 case, though the motivation for that shift is kept brief. The approximation arguments look careful and self-contained. A referee might still want a bit more on how the constants behave or how the general position condition carries over exactly, but nothing in the core equivalence appears to rest on a gap. This is the kind of paper that interests people already working on Nagata-type questions or on using pluripotential theory for algebraic problems. It gives them a different language without lowering the difficulty of the original statement. I would send it for peer review so experts can check the details of the approximations.

Referee Report

0 major / 2 minor

Summary. The manuscript formulates new transcendental versions of the Nagata conjecture in Cn, obtained from pluripotential theory, and proves their equivalence to a version of the Nagata conjecture in Cn. The equivalence is established in both directions by approximating algebraic data (curves with prescribed multiplicities) by plurisubharmonic functions and conversely, under the stated hypotheses on the points and multiplicities.

Significance. If the equivalences hold, the work supplies a concrete bridge between the classical Nagata problem and pluripotential theory, allowing potential transfer of analytic techniques to an algebraic conjecture that remains open for non-square r ≥ 10. The explicit two-way approximation argument between algebraic and plurisubharmonic data is a clear technical contribution.

minor comments (2)

[Introduction] The introduction would benefit from a short paragraph situating the new Cn statements relative to the original Nagata conjecture in P2 and to known partial results for square r.
[Section 2] Notation for the plurisubharmonic functions and the associated Monge-Ampère measures could be collected in a single preliminary subsection for easier reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately summarizes the main contributions.

Circularity Check

0 steps flagged

No significant circularity; equivalence derived explicitly via approximation

full rationale

The paper's central claim is the formulation of pluripotential-theoretic versions asserted to be equivalent to a Cn extension of the Nagata conjecture. The equivalence is established by explicit bidirectional approximation arguments between algebraic curves/multiplicities and plurisubharmonic functions, relying on standard pluripotential theory techniques under stated hypotheses. No step reduces by definition to its own inputs, no fitted parameters are relabeled as predictions, and no load-bearing premise rests on self-citation chains. The derivation remains self-contained with independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities used in the formulations.

pith-pipeline@v0.9.0 · 5668 in / 1002 out tokens · 30522 ms · 2026-05-25T19:05:03.153142+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We formulate new transcendental versions of this conjecture coming from pluripotential theory and which are equivalent to a version in Cn of the Nagata Conjecture. ... ν(g∞,O)=Ω(S) and (ddcg∞)^n=0 in B(O,1)∖{O}. ... If Ω(S)=|S|^{1/n} then ... g∞(z)=|S|^{1/n} ln||z||
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Conjecture (P1). ... gB(O,1)(tS,.) converges ... to r^{1/n} gB(O,1)(O,.). Conjecture (A1). Weak Version ... Ω(S)=r^{1/n}.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

Bedford and B.A

E. Bedford and B.A. Taylor. The D irichlet problem for a complex M onge- A mp\` e re equation. Invent. Math. , 37(1):1--44, 1976

work page 1976
[2]

Bedford and B.A

E. Bedford and B.A. Taylor. A new capacity for plurisubharmonic functions. Acta Math. , 149(1-2):1--40, 1982

work page 1982
[3]

Chudnovsky

G.V. Chudnovsky. Singular points on complex hypersurfaces and multidimensional schwarz lemma. Seminar on Number Theory, Paris 1979/80 , Birkhauser:29--69, 1981

work page 1979
[4]

Coman and S

D. Coman and S. Nivoche. Plurisubharmonic functions with singularities and affine invariants for finite sets in c ^n . Math. Ann. , 322:317--332, 2002

work page 2002
[5]

D. Coman. Entire pluricomplex G reen functions and L elong numbers of projective currents. Proc. Amer. Math. Soc. , 134(7):1927--1935, 2006

work page 1927
[6]

Demailly

J.-P. Demailly. Mesures de M onge- A mp \`e re et caract\'erisation g\'eom\'etrique des vari\'et\'es alg\'ebriques affines. M\' e m. Soc. Math. France , 19:1--125, 1985

work page 1985
[7]

Demailly

J.-P. Demailly. Mesures de M onge- A mp \`e re et mesures pluriharmoniques. Math. Z. , 194(4):519--564, 1987

work page 1987
[8]

Demailly

J.-P. Demailly. Singular H ermitian metrics on positive line bundles. Complex algebraic varieties ( B ayreuth, 1990), Lecture Notes in Math. 1507 , 87-104, 1992

work page 1990
[9]

Demailly

J.-P. Demailly. M onge- A mp\`ere operators, L elong numbers and intersection theory. C omplex analysis and geometry, Univ. Ser. Math., Plenum, New York , 115-193, 1993

work page 1993
[10]

Multiplier ideal sheaves and analytic methods in algebraic geometry

Jean-Pierre Demailly. Multiplier ideal sheaves and analytic methods in algebraic geometry. School on V anishing T heorems and E ffective R esults in A lgebraic G eometry ( T rieste, 2000) , 6:1--148, 2001

work page 2000
[11]

Nguyen Quang Dieu and P. J. Thomas. Convergence of multipole green functions. Indiana Univ. Math. J. , 65(1):223--241, 2016

work page 2016
[12]

L. Evain. On the postulation of s^d fat points in p ^d . J. Algebra , 285:516--530, 2005

work page 2005
[13]

Harbourne

B. Harbourne. On nagata's conjecture. Journal of Algebra , 236:692--702, 2001

work page 2001
[14]

H o rmander

L. H o rmander. Notions of convexity , volume 127 of Progress in Mathematics . Birkh\" a user Boston, Inc., Boston, MA, 1994

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[15]

Discrete behavior of S eshadri constants on surfaces

Brian Harbourne and Joaquim Ro\'e. Discrete behavior of S eshadri constants on surfaces. J. Pure Appl. Algebra , 212(3):616--627, 2008

work page 2008
[16]

Iarrobino

A. Iarrobino. Inverse system of a symbolic power. iii. thin algebras and fat points. Composito Math. , 108(3):319--356, 1997

work page 1997
[17]

M. Klimek. Extremal plurisubharmonic functions and invariant pseudodistances. Bull. Soc. Math. France , 113:231--240, 1985

work page 1985
[18]

P. Lelong. Notions capacitaires et fonctions de G reen pluricomplexes dans les espaces de B anach. C.R.A.S. Paris S\'er. I Math. , 305(3):71--76, 1987

work page 1987
[19]

P. Lelong. Fonction de G reen pluricomplexe et lemmes de S chwarz dans les espaces de B anach. J. Maths. Pures et Appl. , 68:319--347, 1989

work page 1989
[20]

Lemmes de S chwarz en plusieurs variables et applications arithm\'etiques

Jean-Charles Moreau. Lemmes de S chwarz en plusieurs variables et applications arithm\'etiques. S\'eminaire P ierre L elong- H enri S koda ( A nalyse). A nn\'ees 1978/79 ( F rench) , 822:174--190, 1980

work page 1978
[21]

Magnusson, A

J. Magnusson, A. Rashkovskii, R. Sigurdsson, and P. J. Thomas. Limits of multipole pluricomplex G reen functions. Intern. J. Math. , 23(6):38--39, 2012

work page 2012
[22]

M. Nagata. On the fourteenth problem of H ilbert. Proc. Intern. Congress Math. , Cambridge Univ. Press, New York, 1960:459--462, 1958

work page 1960
[23]

M. Nagata. On the 14-th problem of H ilbert. Amer. J. Math. , 81:766--772, 1959

work page 1959
[24]

M. Nagata. On rational surfaces, ii. Mem. Coll. Sci. Univ. Kyoto, Ser. A Math , 33:271--293, 1960

work page 1960
[25]

S. Nivoche. The pluricomplex G reen function, capacitative notions, and approximation problems in c ^n . Indiana Univ. Math. J. , 44(2):489--510, 1995

work page 1995
[26]

Rashkovskii and P

A. Rashkovskii and P. J. Thomas. Powers of ideals and convergence of green functions with colliding poles. Int. Math. Res. Not. IMRN , 2014:1253--1272, 2014

work page 2014
[27]

A bound for S eshadri constants on P ^2

Halszka Tutaj-Gasi\'nska. A bound for S eshadri constants on P ^2 . Math. Nachr. , 257:108--116, 2003

work page 2003
[28]

Waldschmidt

M. Waldschmidt. Propri\'et\'es arithm\'etiques de fonctions de plusieurs variables. ii. (french) S \'eminaire P ierre L elong ( A nalyse) ann\'ee 1975/76. Lecture Notes in Math., Springer, Berlin , 578:108--135, 1977

work page 1975
[29]

Waldschmidt

M. Waldschmidt. Nombres transcendants et groupes alg\'ebriques , volume 218 pp. of Ast\'erisque No. 69-70 . Second edition, 1987

work page 1987
[30]

Curves in P ^2 and symplectic packings

Geng Xu. Curves in P ^2 and symplectic packings. Math. Ann. , 299(4):609--613, 1994

work page 1994

[1] [1]

Bedford and B.A

E. Bedford and B.A. Taylor. The D irichlet problem for a complex M onge- A mp\` e re equation. Invent. Math. , 37(1):1--44, 1976

work page 1976

[2] [2]

Bedford and B.A

E. Bedford and B.A. Taylor. A new capacity for plurisubharmonic functions. Acta Math. , 149(1-2):1--40, 1982

work page 1982

[3] [3]

Chudnovsky

G.V. Chudnovsky. Singular points on complex hypersurfaces and multidimensional schwarz lemma. Seminar on Number Theory, Paris 1979/80 , Birkhauser:29--69, 1981

work page 1979

[4] [4]

Coman and S

D. Coman and S. Nivoche. Plurisubharmonic functions with singularities and affine invariants for finite sets in c ^n . Math. Ann. , 322:317--332, 2002

work page 2002

[5] [5]

D. Coman. Entire pluricomplex G reen functions and L elong numbers of projective currents. Proc. Amer. Math. Soc. , 134(7):1927--1935, 2006

work page 1927

[6] [6]

Demailly

J.-P. Demailly. Mesures de M onge- A mp \`e re et caract\'erisation g\'eom\'etrique des vari\'et\'es alg\'ebriques affines. M\' e m. Soc. Math. France , 19:1--125, 1985

work page 1985

[7] [7]

Demailly

J.-P. Demailly. Mesures de M onge- A mp \`e re et mesures pluriharmoniques. Math. Z. , 194(4):519--564, 1987

work page 1987

[8] [8]

Demailly

J.-P. Demailly. Singular H ermitian metrics on positive line bundles. Complex algebraic varieties ( B ayreuth, 1990), Lecture Notes in Math. 1507 , 87-104, 1992

work page 1990

[9] [9]

Demailly

J.-P. Demailly. M onge- A mp\`ere operators, L elong numbers and intersection theory. C omplex analysis and geometry, Univ. Ser. Math., Plenum, New York , 115-193, 1993

work page 1993

[10] [10]

Multiplier ideal sheaves and analytic methods in algebraic geometry

Jean-Pierre Demailly. Multiplier ideal sheaves and analytic methods in algebraic geometry. School on V anishing T heorems and E ffective R esults in A lgebraic G eometry ( T rieste, 2000) , 6:1--148, 2001

work page 2000

[11] [11]

Nguyen Quang Dieu and P. J. Thomas. Convergence of multipole green functions. Indiana Univ. Math. J. , 65(1):223--241, 2016

work page 2016

[12] [12]

L. Evain. On the postulation of s^d fat points in p ^d . J. Algebra , 285:516--530, 2005

work page 2005

[13] [13]

Harbourne

B. Harbourne. On nagata's conjecture. Journal of Algebra , 236:692--702, 2001

work page 2001

[14] [14]

H o rmander

L. H o rmander. Notions of convexity , volume 127 of Progress in Mathematics . Birkh\" a user Boston, Inc., Boston, MA, 1994

work page 1994

[15] [15]

Discrete behavior of S eshadri constants on surfaces

Brian Harbourne and Joaquim Ro\'e. Discrete behavior of S eshadri constants on surfaces. J. Pure Appl. Algebra , 212(3):616--627, 2008

work page 2008

[16] [16]

Iarrobino

A. Iarrobino. Inverse system of a symbolic power. iii. thin algebras and fat points. Composito Math. , 108(3):319--356, 1997

work page 1997

[17] [17]

M. Klimek. Extremal plurisubharmonic functions and invariant pseudodistances. Bull. Soc. Math. France , 113:231--240, 1985

work page 1985

[18] [18]

P. Lelong. Notions capacitaires et fonctions de G reen pluricomplexes dans les espaces de B anach. C.R.A.S. Paris S\'er. I Math. , 305(3):71--76, 1987

work page 1987

[19] [19]

P. Lelong. Fonction de G reen pluricomplexe et lemmes de S chwarz dans les espaces de B anach. J. Maths. Pures et Appl. , 68:319--347, 1989

work page 1989

[20] [20]

Lemmes de S chwarz en plusieurs variables et applications arithm\'etiques

Jean-Charles Moreau. Lemmes de S chwarz en plusieurs variables et applications arithm\'etiques. S\'eminaire P ierre L elong- H enri S koda ( A nalyse). A nn\'ees 1978/79 ( F rench) , 822:174--190, 1980

work page 1978

[21] [21]

Magnusson, A

J. Magnusson, A. Rashkovskii, R. Sigurdsson, and P. J. Thomas. Limits of multipole pluricomplex G reen functions. Intern. J. Math. , 23(6):38--39, 2012

work page 2012

[22] [22]

M. Nagata. On the fourteenth problem of H ilbert. Proc. Intern. Congress Math. , Cambridge Univ. Press, New York, 1960:459--462, 1958

work page 1960

[23] [23]

M. Nagata. On the 14-th problem of H ilbert. Amer. J. Math. , 81:766--772, 1959

work page 1959

[24] [24]

M. Nagata. On rational surfaces, ii. Mem. Coll. Sci. Univ. Kyoto, Ser. A Math , 33:271--293, 1960

work page 1960

[25] [25]

S. Nivoche. The pluricomplex G reen function, capacitative notions, and approximation problems in c ^n . Indiana Univ. Math. J. , 44(2):489--510, 1995

work page 1995

[26] [26]

Rashkovskii and P

A. Rashkovskii and P. J. Thomas. Powers of ideals and convergence of green functions with colliding poles. Int. Math. Res. Not. IMRN , 2014:1253--1272, 2014

work page 2014

[27] [27]

A bound for S eshadri constants on P ^2

Halszka Tutaj-Gasi\'nska. A bound for S eshadri constants on P ^2 . Math. Nachr. , 257:108--116, 2003

work page 2003

[28] [28]

Waldschmidt

M. Waldschmidt. Propri\'et\'es arithm\'etiques de fonctions de plusieurs variables. ii. (french) S \'eminaire P ierre L elong ( A nalyse) ann\'ee 1975/76. Lecture Notes in Math., Springer, Berlin , 578:108--135, 1977

work page 1975

[29] [29]

Waldschmidt

M. Waldschmidt. Nombres transcendants et groupes alg\'ebriques , volume 218 pp. of Ast\'erisque No. 69-70 . Second edition, 1987

work page 1987

[30] [30]

Curves in P ^2 and symplectic packings

Geng Xu. Curves in P ^2 and symplectic packings. Math. Ann. , 299(4):609--613, 1994

work page 1994