Skoda division of line bundle sections and pseudo-division

TL;DR

This paper introduces a highly general Skoda division theorem for line bundle sections, revisits related Nullstellensatz results, and proposes pseudo-division as a new tool to improve finite generation proofs.

Contribution

It develops a notion of pseudo-division to enhance the Skoda division framework and applies it to achieve vanishing order 1 pseudo-division for ample line bundles.

Findings

A general Skoda division theorem derived from Varolin's theorem.

Pseudo-division can replace traditional division in finite generation arguments.

Established vanishing order 1 pseudo-division for ample line bundles.

Abstract

We first present a Skoda type division theorem for holomorphic sections of line bundles on a projective variety which is essentially the most general, compared to previous ones. It is derived from Varolin's theorem as a corollary. Then we revisit Geometric Effective Nullstellensatz and observe that even this general Skoda division is far from sufficient to yield stronger Geometric Effective Nullstellensatz such as `vanishing order $1$ division', which could be used for finite generation of section rings by the basic finite generation lemma. To resolve this problem, we develop a notion of pseudo-division and show that it can replace the usual division in the finite generation lemma. We also give a vanishing order 1 pseudo-division result when the line bundle is ample.