Seshadri constants via Lelong numbers

TL;DR

This paper explores a new method to compute Seshadri constants using Lelong numbers, applying it to various algebraic geometry problems including blown-up products of curves and Nagata's conjecture.

Contribution

It introduces a novel approach linking Lelong numbers with Seshadri constants, providing new formulas and applications in algebraic geometry.

Findings

Computed Seshadri constants on blown-up products of curves.

Disproved a conjecture on maximal rationally connected quotients.

Proposed a new approach to Nagata's conjecture.

Abstract

One of Demailly's characterizations of Seshadri constants on ample line bundles works with Lelong numbers of certain positive singular hermitian metrics. In this note sections of multiples of the line bundle are used to produce such metrics and then to deduce another formula for Seshadri constants. It is applied to compute Seshadri constants on blown up products of curves, to disprove a conjectured characterization of maximal rationally connected quotients and to introduce a new approach to Nagata's conjecture.