On blow-analytic equivalence of embedded curve singularities. (English) Zbl 0899.32002

Kobayashi, Masanori; Kuo, Tzee-Char

Fukuda, T. (ed.) et al., Real analytic and algebraic singularities. Harlow: Longman. Pitman Res. Notes Math. Ser. 381, 30-37 (1998).

A homeomorphism \(h:({\mathbb{R}}^2,0) \to ({\mathbb{R}}^2,0)\) is blow-analytic if there exists a modification \(\pi: (X,E)\to ({\mathbb{R}}^2,0)\) such that the composite \(h\circ \pi\) is analytic and if the corresponding condition for \(h^{-1}\) holds.
By a result of T. Fukui [Compos. Math. 105, No. 1, 95-108 (1997; Zbl 0873.32008)] the multiplicity of a germ of a curve is a blow-analytic invariant. This is no longer the case if one considers only the underlying topological spaces and ignores analytic structures. The main result of this paper is that the germ \(C\) of any unibranch real plane curve singularity can be transformed into the germ of a smooth line by a blow-analytic homeomorphism. The proof is based on the fact that two embedded resolutions are equivalent if their weighted graphs are the same modulo 2.
For the entire collection see [Zbl 0882.00014].

Reviewer: J.Stevens (Göteborg)

MSC:

32C07	Real-analytic sets, complex Nash functions
32S15	Equisingularity (topological and analytic)

Keywords:

real analytic curve; blow-analytic equivalence; embedded curve singularities