The Manin-Peyre conjecture for smooth spherical Fano threefolds

TL;DR

This paper proves the Manin-Peyre conjecture for a class of smooth spherical Fano threefolds, extending previous results and addressing unique structural challenges such as fractional exponents and exceptional subsets.

Contribution

It establishes the conjecture for semisimple rank one, type N spherical Fano threefolds, introducing new structural insights and methods to handle irregularities and fractional height conditions.

Findings

Confirmed Manin-Peyre conjecture for type N spherical Fano threefolds

Addressed structural novelties including fractional exponents

Identified the need to exclude thin subsets for accurate counting

Abstract

The Manin-Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The case N features a number of structural novelties; most notably, one may lose regularity of the ambient toric variety, the height conditions may contain fractional exponents, and it may be necessary to exclude a thin subset with exceptionally many rational points from the count, as otherwise Manin's conjecture in its original form would turn out to be incorrect.