Approximating rational points on horospherical varieties

TL;DR

Under Vojta's conjecture, the paper constructs curves on horospherical varieties through rational points that approximate these points more closely than Zariski dense sequences, advancing understanding of rational point distribution.

Contribution

It proves a weakened form of McKinnon's conjecture for horospherical varieties using minimal model program techniques.

Findings

Constructed curves through rational points with superior approximation properties.

Proved a weakened conjecture of McKinnon in the horospherical setting.

Applied methods to varieties with terminal singularities.

Abstract

Let $X$ be a smooth projective split horospherical variety over a number field $k$ and $x\in X(k)$. Contingent on Vojta's conjecture, we construct a curve $C$ through $x$ such that (in a precise sense) rational points on $C$ approximate $x$ better than any Zariski dense sequence of rational points. This proves a weakening of a conjecture of McKinnon in the horospherical case. Our results make use of the minimal model program and apply as well to $\mathbb{Q}$-factorial horospherical varieties with terminal singularities.