Multiplicity and the pull-back problem

TL;DR

This paper generalizes a formula related to intersection indices and applies it to prove that under certain conditions, the smoothness of a preimage under a finite holomorphic map implies the smoothness of the target variety.

Contribution

It extends Spodzieja's intersection index formula to the isolated improper Achilles-Tworzewski-Winiarski case and provides a new proof of a smoothness transfer result for finite holomorphic maps.

Findings

Generalized intersection index formula for isolated improper cases

Proved that smooth preimages imply smooth targets under specific conditions

Simplified proof of a known smoothness transfer theorem

Abstract

We discuss a formula of S. Spodzieja and generalize it for the isolated improper Achilles-Tworzewski-Winiarski intersection index. As an application we give a simple proof of a result of P. Ebenfelt and L. Rothschild: if $F\colon (\mathbb{C}^m,0)\to (\mathbb{C}^m,0)$ is a finite holomorphic map, $W$ a germ of a complex variety at zero such that $F^{-1}(W)$ is a smooth germ and the Jacobian of $F$ does not vanish identically on it, then $W$ is smooth too.