Semi-integral points of bounded height on toric varieties

TL;DR

This paper establishes asymptotic formulas for semi-integral points of bounded height on toric varieties, confirming a Manin-type conjecture for specific smooth and singular cases with predicted leading constants.

Contribution

It provides the first proof of asymptotics for semi-integral points on toric varieties and verifies a conjecture with precise leading constants for certain orbifolds.

Findings

Asymptotic formulas for semi-integral points established

Verification of Manin-type conjecture for specific toric orbifolds

Leading constants match predictions in tested cases

Abstract

We prove asymptotics for semi-integral points of bounded height on toric varieties. We verify the Manin-type conjecture of Pieropan, Smeets, Tanimoto and V\'arilly-Alvarado for smooth and certain singular toric orbifolds upon replacing the leading constant with the one predicted by Chow, Loughran, Takloo-Bighash and Tanimoto.