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The ultrametric space of plane branches

AuthorsAbío, Ignasi; Alberich-Carramiñana, Maria ; González-Alonso, Víctor
KeywordsUltrametric distance
Equisingularity class
Plane branch
Issue Date2011
PublisherTaylor & Francis
CitationCommunications in Algebra 39(11): 4206-4220 (2011)
AbstractWe 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.
Publisher version (URL)http://dx.doi.org/10.1080/00927872.2010.521934
Identifiersdoi: 10.1080/00927872.2010.521934
issn: 0092-7872
e-issn: 1532-4125
Appears in Collections:(IRII) Artículos
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