Rational approximations on toric varieties

Huang, Zhizhong

doi:10.2140/ant.2021.15.461

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Rational approximations on toric varieties

Zhizhong Huang

Vol. 15 (2021), No. 2, 461–512

DOI: 10.2140/ant.2021.15.461

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

Mathematical Subject Classification

Primary: 14G05

Secondary: 14M25, 11J99

Milestones

Received: 1 January 2020

Revised: 27 July 2020

Accepted: 24 August 2020

Published: 7 April 2021

Authors

Zhizhong Huang
Institut für Algebra, Zahlentheorie und Diskrete Mathematik Leibniz Universität Hannover Hannover Germany

Vol. 15, No. 2, 2021