Rational approximations on toric varieties

TL;DR

This paper investigates how well rational points can approximate fixed points on split toric varieties, confirming a conjecture that minimal degree rational curves provide the best approximations under certain geometric conditions.

Contribution

It proves that minimal degree rational curves passing through a fixed point yield optimal approximations on split toric varieties, confirming McKinnon's conjecture.

Findings

Best approximations are achieved on rational curves of minimal degree.

The minimal degree curves correspond to centred primitive collections of the fan.

The results are established under specific geometric hypotheses.

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-Roth's work) can be achieved on rational curves passing through the fixed point of minimal degree, confirming a conjecture of McKinnon. These curves correspond, according to Batyrev's terminology, to the centred primitive collections of the fan.