Details
| Original language | English |
|---|---|
| Pages (from-to) | 461-512 |
| Number of pages | 52 |
| Journal | Algebra and Number Theory |
| Volume | 15 |
| Issue number | 2 |
| Publication status | Published - 7 Apr 2021 |
Abstract
Using the universal torsor method due to Salberger, we study the approximation of a general fixed point by rational points on split toric varieties. We prove that under certain geometric hypothesis the best approximations (in the sense of McKinnon and Roth’s work) can be achieved on rational curves passing through the fixed point of minimal degree, confirming a conjecture of McKinnon. These curves are also minimal in the sense of deformation theory, and they correspond, according to Batyrev’s terminology, to the centred primitive collections of the structural fan.
Keywords
- Diophantine approximation of rational points, Toric varieties, Universal torsors
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Algebra and Number Theory, Vol. 15, No. 2, 07.04.2021, p. 461-512.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Rational approximations on toric varieties
AU - Huang, Zhizhong
N1 - Funding Information: This paper grew out of part of my Ph.D. thesis realised at Université Grenoble Alpes. I would like to thank Emmanuel Peyre for constant encouragement over the past few years, and I’m grateful to David McKinnon for his interest. The idea of considering primitive collections was brought up by Michel Brion, to whom I address my gratitude. A special thanks goes to the anonymous referee for numerous suggestions which lead to significant improvements in the exposition. When working on this project the author was partly supported by the project ANR GARDIO, by a Riemann fellowship and by grant DE 1646/4-2 of the Deutsche Forschungsgemeinschaft.
PY - 2021/4/7
Y1 - 2021/4/7
N2 - Using the universal torsor method due to Salberger, we study the approximation of a general fixed point by rational points on split toric varieties. We prove that under certain geometric hypothesis the best approximations (in the sense of McKinnon and Roth’s work) can be achieved on rational curves passing through the fixed point of minimal degree, confirming a conjecture of McKinnon. These curves are also minimal in the sense of deformation theory, and they correspond, according to Batyrev’s terminology, to the centred primitive collections of the structural fan.
AB - Using the universal torsor method due to Salberger, we study the approximation of a general fixed point by rational points on split toric varieties. We prove that under certain geometric hypothesis the best approximations (in the sense of McKinnon and Roth’s work) can be achieved on rational curves passing through the fixed point of minimal degree, confirming a conjecture of McKinnon. These curves are also minimal in the sense of deformation theory, and they correspond, according to Batyrev’s terminology, to the centred primitive collections of the structural fan.
KW - Diophantine approximation of rational points
KW - Toric varieties
KW - Universal torsors
UR - http://www.scopus.com/inward/record.url?scp=85105148143&partnerID=8YFLogxK
U2 - 10.2140/ant.2021.15.461
DO - 10.2140/ant.2021.15.461
M3 - Article
AN - SCOPUS:85105148143
VL - 15
SP - 461
EP - 512
JO - Algebra and Number Theory
JF - Algebra and Number Theory
SN - 1937-0652
IS - 2
ER -