Characterization of negative line bundles whose Grauert blow-down are quadratic transforms

TL;DR

This paper characterizes when the Grauert blow-down of a negative line bundle over a compact complex space results in a quadratic transform, based on ampleness and global generation conditions of certain line bundle powers.

Contribution

It provides a precise criterion linking the properties of a negative line bundle to the nature of its Grauert blow-down as a quadratic transform.

Findings

Grauert blow-down is quadratic iff $k_0L^*$ is very ample and $(k_0+1)L^*$ is globally generated.

The characterization depends on the initial order of the dual line bundle.

Establishes a clear geometric condition for quadratic transforms in complex geometry.

Abstract

We show that the Grauert blow-down of a holomorphic negative line bundle $L$ over a compact complex space is a quadratic transform if and only if $k_0L^*$ is very ample and $(k_0+1)L^*$ is globally generated, where $k_0$ is the initial order of $L^*$, namely, the minimal integer such that $k_0^*$ has nontrivial holomorphic section.