Birational Nevanlinna Constants, Beta Constants, and Diophantine Approximation to Closed Subschemes
Paul Vojta1
1 Department of Mathematics University of California 970 Evans Hall #3840 Berkeley, CA 94720-3840 USA
Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 17-61.

In an earlier paper (joint with Min Ru), we proved a result on diophantine approximation to Cartier divisors, extending a 2011 result of P. Autissier. This was recently extended to certain closed subschemes (in place of divisors) by Ru and Wang. In this paper we extend this result to a broader class of closed subschemes. We also show that some notions of β(ℒ,D) coincide, and that they can all be evaluated as limits.

Dans un article antérieur (en commun avec Min Ru), nous avons prouvé un résultat sur l’approximation diophantienne relativement aux diviseurs de Cartier, en généralisant un résultat de 2011 de P. Autissier. Cela a été récemment étendu à certains sous-schémas fermés (à la place de diviseurs) par Ru et Wang. Dans cet article, nous étendons ce résultat à une classe de sous-schémas fermés plus large. Nous montrons également que certaines notions de β(ℒ,D) coïncident, et qu’elles peuvent toutes être évaluées comme des limites.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1237
Classification: 11J97, 32H30
Keywords: Nevanlinna constant, b-divisors, closed subscheme
Paul Vojta 1

1 Department of Mathematics University of California 970 Evans Hall #3840 Berkeley, CA 94720-3840 USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{JTNB_2023__35_1_17_0,
     author = {Paul Vojta},
     title = {Birational {Nevanlinna} {Constants,} {Beta} {Constants,} and {Diophantine} {Approximation} to {Closed} {Subschemes}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {17--61},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {35},
     number = {1},
     year = {2023},
     doi = {10.5802/jtnb.1237},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1237/}
}
TY  - JOUR
AU  - Paul Vojta
TI  - Birational Nevanlinna Constants, Beta Constants, and Diophantine Approximation to Closed Subschemes
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2023
SP  - 17
EP  - 61
VL  - 35
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1237/
DO  - 10.5802/jtnb.1237
LA  - en
ID  - JTNB_2023__35_1_17_0
ER  - 
%0 Journal Article
%A Paul Vojta
%T Birational Nevanlinna Constants, Beta Constants, and Diophantine Approximation to Closed Subschemes
%J Journal de théorie des nombres de Bordeaux
%D 2023
%P 17-61
%V 35
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1237/
%R 10.5802/jtnb.1237
%G en
%F JTNB_2023__35_1_17_0
Paul Vojta. Birational Nevanlinna Constants, Beta Constants, and Diophantine Approximation to Closed Subschemes. Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 17-61. doi : 10.5802/jtnb.1237. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1237/

[1] Pascal Autissier Sur la non-densité des points entiers, Duke Math. J., Volume 158 (2011) no. 1, pp. 13-27 | MR | Zbl

[2] Robin Hartshorne Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | DOI

[3] Gordon Heier; Aaron Levin A generalized Schmidt subspace theorem for closed subschemes, Am. J. Math., Volume 143 (2021) no. 1, pp. 213-226 | DOI | MR | Zbl

[4] P. Jothilingam; T. Duraivel; Sibnath Bose Regular sequences of ideals, Beitr. Algebra Geom., Volume 52 (2011) no. 2, pp. 479-485 | DOI | Zbl

[5] Serge Lang Fundamentals of Diophantine geometry, Springer, 1983 | DOI

[6] Robert Lazarsfeld Positivity in Algebraic Geometry I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete, 48, 49, Springer, 2004

[7] Min Ru A defect relation for holomorphic curves intersecting general divisors in projective varieties, J. Geom. Anal., Volume 26 (2016) no. 4, pp. 2751-2776 | MR | Zbl

[8] Min Ru On a general Diophantine inequality, Funct. Approximatio, Comment. Math., Volume 56 (2017) no. 2, pp. 143-163 | MR | Zbl

[9] Min Ru; Paul Vojta A birational Nevanlinna constant and its consequences, Am. J. Math., Volume 142 (2020) no. 3, pp. 957-991 | MR | Zbl

[10] Min Ru; Paul Vojta An Evertse–Ferretti Nevanlinna constant and its consequences, Monatsh. Math., Volume 196 (2021), pp. 305-334 | DOI | MR | Zbl

[11] Min Ru; Julie Tzu-Yueh Wang A subspace theorem for subvarieties, Algebra Number Theory, Volume 11 (2017) no. 10, pp. 2323-2337 | MR | Zbl

[12] Min Ru; Julie Tzu-Yueh Wang The Ru–Vojta result for subvarieties, Int. J. Number Theory, Volume 18 (2022) no. 1, pp. 61-74 | MR | Zbl

[13] Joseph H. Silverman Arithmetic distance functions and height functions in diophantine geometry, Math. Ann., Volume 279 (1987) no. 1-2, pp. 193-216 | DOI | MR | Zbl

[14] The Stacks Project Authors Stacks Project, 2020 (http://stacks.math.columbia.edu)

[15] Paul Vojta A refinement of Schmidt’s Subspace Theorem, Am. J. Math., Volume 111 (1989) no. 3, pp. 489-518 | DOI | MR | Zbl

[16] Paul Vojta Diophantine approximation and Nevanlinna theory, Arithmetic geometry (Lecture Notes in Mathematics), Volume 2009, Springer, 2011, pp. 111-224 | DOI | MR

[17] Paul Vojta Roth’s Theorem over arithmetic function fields, Algebra Number Theory, Volume 15 (2021) no. 8, pp. 1943-2017 | DOI | MR | Zbl

[18] Katsutoshi Yamanoi Algebro-geometric version of Nevanlinna’s lemma on logarithmic derivative and applications, Nagoya Math. J., Volume 173 (2004), pp. 23-63 | DOI | MR | Zbl

Cited by Sources: