Vol. 15, No. 2, 2021

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Rational approximations on toric varieties

Zhizhong Huang

Vol. 15 (2021), No. 2, 461–512

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.

Diophantine approximation of rational points, toric varieties, universal torsors
Mathematical Subject Classification
Primary: 14G05
Secondary: 14M25, 11J99
Received: 1 January 2020
Revised: 27 July 2020
Accepted: 24 August 2020
Published: 7 April 2021
Zhizhong Huang
Institut für Algebra, Zahlentheorie und Diskrete Mathematik
Leibniz Universität Hannover