The Ultrametric Space of Plane Branches

We study properties of the space of irreducible germs of plane curves (branches), seen as an ultrametric space. We provide various geometrical methods to measure the distance between two branches and to compare distances between branches, in terms of topological invariants of the singularity which c...

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Bibliographic Details
Published in:Communications in algebra Vol. 39; no. 11; pp. 4206 - 4220
Main Authors: Abío, Ignasi, Alberich-Carramiñana, Maria, González-Alonso, Víctor
Format: Journal Article Publication
Language:English
Published: Taylor & Francis Group 01.11.2011
Subjects:
14 Algebraic geometry
14H Curves
32 Several complex variables and analytic spaces
32S Singularities
Classificació AMS
Corbes planes
Curves, Plane
Equisingularity class
Matemàtiques i estadística
Plane branch
Primary 14H20
Secondary 32S05, 14C17
Singularitats (Matemàtica)
Ultrametric distance
Àrees temàtiques de la UPC
ISSN:0092-7872, 1532-4125
Online Access:Get full text
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Summary:We study properties of the space of irreducible germs of plane curves (branches), seen as an ultrametric space. We provide various geometrical methods to measure the distance between two branches and to compare distances between branches, in terms of topological invariants of the singularity which comprises some of the branches. We show that, in spite of being very close to the notion of intersection multiplicity between two germs, this notion of distance behaves very differently. For instance, any value in [0, 1] ∩ ℚ is attained as the distance between a fixed branch and some other branch, in contrast with the fact that the semigroup of the fixed branch has gaps. We also present results that lead to interpret this distance as a sort of geometric distance between the topological equivalence or equisingularity classes of branches.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2010.521934