Lipschitz geometry of complex surface germs via inner rates of primary ideals

TL;DR

This paper links the Lipschitz geometry of complex surface germs to algebraic invariants called inner rates of primary ideals, showing how these invariants determine the outer Lipschitz structure and constructing diverse examples with the same normalization.

Contribution

It introduces a method to associate continuous functions to primary ideals that encode Lipschitz geometry, and constructs families of surface germs with identical normalization but different Lipschitz types.

Findings

The function $ ext{I}_I$ is determined by the outer Lipschitz geometry.

Existence of surface germs with the same normalization but different Lipschitz types.

Construction of infinite families of surface germs with distinct Lipschitz geometries.

Abstract

Let $(X, 0)$ be a normal complex surface germ embedded in $(\mathbb{C}^n, 0)$, and denote by $\mathfrak{m}$ the maximal ideal of the local ring $\mathcal{O}_{X,0}$. In this paper, we associate to each $\mathfrak{m}$-primary ideal $I$ of $\mathcal{O}_{X,0}$ a continuous function $\mathcal{I}_I$ defined on the set of positive (suitably normalized) semivaluations of $\mathcal{O}_{X,0}$. We prove that the function $\mathcal{I}_{\mathfrak{m}}$ is determined by the outer Lipschitz geometry of the surface $(X, 0)$. We further demonstrate that for each $\mathfrak{m}$-primary ideal $I$, there exists a complex surface germ $(X_I, 0)$ with an isolated singularity whose normalization is isomorphic to $(X, 0)$ and $\mathcal{I}_I = \mathcal{I}_{\mathfrak{m}_I}$, where $\mathfrak{m}_I$ is the maximal ideal of $\mathcal{O}_{X_I,0}$. Subsequently, we construct an infinite family of complex surface germs with isolated singularities, whose normalizations are isomorphic to $(X,0)$ (in particular, they are homeomorphic to $(X,0)$) but have distinct outer Lipschitz types.