Determinant method and the pseudo-effective threshold

TL;DR

This paper refines the determinant method by providing an upper bound on auxiliary hypersurfaces, linking key constants to the pseudo-effective threshold of line bundles using Arakelov geometry.

Contribution

It introduces a new upper bound for the number of auxiliary hypersurfaces in the determinant method, connecting it to the pseudo-effective threshold via Arakelov geometry.

Findings

Upper bound on auxiliary hypersurfaces established

Key constants determined by pseudo-effective thresholds

Reformulation of Salberger's work using Arakelov geometry

Abstract

In this paper, we will give an upper bound of the number of auxiliary hypersurfaces in the determinant method, which reformulates an unpublished work of Salberger by Arakelov geometry. One of the key constants will be determined by the pseudo-effective threshold of certain line bundles.