Multiple point spaces of finite holomorphic maps

TL;DR

This paper establishes a unique, natural definition for multiple point spaces of finite holomorphic maps, aligning with known cases and providing methods for their computation.

Contribution

It introduces a canonical definition of multiple point spaces for holomorphic maps, extending previous special cases and exploring their properties and computational aspects.

Findings

The definition coincides with known double and multiple point spaces for corank one maps.

Double point spaces can often be computed as zero loci of ideal quotients.

The paper proves uniqueness and naturality of the proposed definition.

Abstract

We show that there exists a unique possible definition, with certain natural properties, of the multiple point space of a holomorphic map between complex manifolds. Our construction coincides with the double point space and the k-th multiple point space for corank one map-germs, due to Mond. We also give some interesting properties of the double point space and prove that in many cases it can be computed as the zero locus of certain quotient of ideals.