Strict germs on normal surface singularities

TL;DR

This paper demonstrates that holomorphic germs of topological degree one between normal surface singularities can be decomposed into a modification and a local isomorphism, with implications for understanding self-maps and singularity types.

Contribution

It provides an alternative proof for the structure of degree-one holomorphic germs using Kato surfaces and valuative dynamics, extending previous results on sandwiched singularities.

Findings

Any degree-one holomorphic germ decomposes into a modification and a local isomorphism.

When the germ is a self-map, the singularity is sandwiched.

The proof employs Kato surfaces and valuative dynamics techniques.

Abstract

We show that any holomorphic germ $f \colon (X,x_0) \to (Y,y_0)$ of topological degree $1$ between normal surface singularities can be written as $f=\pi \circ \sigma$, where $\pi \colon Y' \to (Y,y_0)$ is a modification and $\sigma \colon (X,x_0) \to (Y',y_1)$ is a local isomorphism sending $x_0$ to a point $y_1 \in \pi^{-1}(y_0)$. A result by Fantini, Favre and myself guarantees that when $f$ is a selfmap, then $(X,x_0)$ is a sandwiched singularity. We give here an alternative proof based on the construction of the associated Kato surfaces, and valuative dynamics.