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On the blow-analytic equivalence of tribranched plane curves. (English) Zbl 1343.32005

Summary: We prove the finiteness of the number of blow-analytic equivalence classes of embedded plane curve germs for any fixed number of branches and for any fixed value of \(\mu'\) – a combinatorial invariant coming from the dual graphs of good resolutions of embedded plane curve singularities. In order to do so, we develop the concept of standard form of a dual graph. We show that, fixed \(\mu'\) in \(\mathbb{N}\), there are only a finite number of standard forms, and to each one of them correspond a finite number of blow-analytic equivalence classes. In the tribranched case, we are able to give an explicit upper bound to the number of graph standard forms. For \(\mu'\leq 2\), we also provide a complete list of standard forms.

MSC:

32C07 Real-analytic sets, complex Nash functions
14P15 Real-analytic and semi-analytic sets
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References:

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