Manin's conjecture for certain spherical threefolds

TL;DR

This paper proves Manin's conjecture for certain spherical threefolds, establishing the asymptotic count of rational points and revealing exceptions to existing conjectural formulas.

Contribution

It extends Manin's conjecture to spherical threefolds with singularities and identifies a counterexample to Batyrev and Tschinkel's conjecture on the leading constant.

Findings

Proved Manin's conjecture for specific spherical threefolds

Identified a family that contradicts Batyrev and Tschinkel's conjecture

Utilized universal torsor method and Cox rings in proofs

Abstract

We prove Manin's conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for a spherical threefold with canonical singularities and two infinite families of spherical threefolds with log terminal singularities. Moreover, we show that one of these families does not satisfy a conjecture of Batyrev and Tschinkel on the leading constant in the asymptotic formula. Our proofs are based on the universal torsor method, using Brion's description of Cox rings of spherical varieties.