Iitaka fibrations and integral points: a family of arbitrarily polarized spherical threefolds

TL;DR

This paper investigates the distribution of integral points on a family of spherical log Fano threefolds, confirming a recent conjecture and advancing understanding of logarithmic Iitaka fibrations.

Contribution

It establishes the asymptotic count of integral points for these threefolds and explores the logarithmic analogue of Iitaka fibrations, which was previously not well-understood.

Findings

Confirmed a conjecture by Santens on integral points.

Determined asymptotic growth of integral points with bounded height.

Provided new insights into logarithmic Iitaka fibrations.

Abstract

Studying Manin's program for a family of spherical log Fano threefolds, we determine the asymptotic number of integral points whose height associated with an arbitrary ample line bundle is bounded. This confirms a recent conjecture by Santens and sheds new light on the logarithmic analogue of Iitaka fibrations, which have not yet been adequately formulated.