Balanced line bundles on Fano varieties

TL;DR

This paper explores the concept of balanced line bundles on Fano varieties, linking geometric invariants to arithmetic properties and rational point distributions, within the framework of the Minimal Model Program.

Contribution

It introduces the notion of balanced line bundles in the context of Fano varieties and the Minimal Model Program, connecting geometric invariants to arithmetic distribution of rational points.

Findings

Balanced line bundles relate to effective divisor cones.

Thresholds with respect to divisor cones influence rational point counts.

Application to classification of Fano threefolds.

Abstract

A conjecture of Batyrev and Manin relates arithmetic properties of varieties with ample anticanonical class to geometric invariants; in particular, counting functions defined by metrized ample line bundles and the corresponding asymptotics of rational points of bounded height are interpreted in terms of cones of effective divisors and certain thresholds with respect to these cones. This framework leads to the notion of balanced line bundles, whose counting functions, conjecturally, capture generic distributions of rational points. We investigate balanced line bundles in the context of the Minimal Model Program, with special regard to the classification of Fano threefolds.