{"title":"Lipschitz geometry of complex surface germs via inner rates of primary ideals","authors":"Yenni Cherik","doi":"arxiv-2407.14265","DOIUrl":null,"url":null,"abstract":"Let $(X, 0)$ be a normal complex surface germ embedded in $(\\mathbb{C}^n,\n0)$, and denote by $\\mathfrak{m}$ the maximal ideal of the local ring\n$\\mathcal{O}_{X,0}$. In this paper, we associate to each $\\mathfrak{m}$-primary\nideal $I$ of $\\mathcal{O}_{X,0}$ a continuous function $\\mathcal{I}_I$ defined\non the set of positive (suitably normalized) semivaluations of\n$\\mathcal{O}_{X,0}$. We prove that the function $\\mathcal{I}_{\\mathfrak{m}}$ is\ndetermined by the outer Lipschitz geometry of the surface $(X, 0)$. We further\ndemonstrate that for each $\\mathfrak{m}$-primary ideal $I$, there exists a\ncomplex surface germ $(X_I, 0)$ with an isolated singularity whose\nnormalization is isomorphic to $(X, 0)$ and $\\mathcal{I}_I =\n\\mathcal{I}_{\\mathfrak{m}_I}$, where $\\mathfrak{m}_I$ is the maximal ideal of\n$\\mathcal{O}_{X_I,0}$. Subsequently, we construct an infinite family of complex\nsurface germs with isolated singularities, whose normalizations are isomorphic\nto $(X,0)$ (in particular, they are homeomorphic to $(X,0)$) but have distinct\nouter Lipschitz types.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"61 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.14265","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.