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Resolving singularities of plane analytic branches with one toric morphism. (English) Zbl 0995.14002
Hauser, H. (ed.) et al., Resolution of singularities. A research textbook in tribute to Oscar Zariski. Based on the courses given at the working week in Obergurgl, Austria, September 7-14, 1997. Basel: Birkhäuser. Prog. Math. 181, 315-340 (2000).
Let \(({\mathcal C},0)\) be an irreducible germ of a complex plane curve. Let \(R\) be the local analytic algebra of \({\mathcal C}\) and \(\nu\) be the unique valuation of \(R\). The graded ring gr\(_{\nu}(R):= \bigoplus_n {I_n\over I_{n+1}}\) where \(I_n:=\{x \in R|\nu(x)\geq n\}\), \(n\in \mathbb{N}\), is the ring of the monomial curve \({\mathcal C}^{\Gamma}\) with the same semigroup \(\Gamma\) as \({\mathcal C}\). There exists a one parameter deformation of \({\mathcal C}^{\Gamma}\) having all its fibers except the special one isomorphic to \({\mathcal C}\). In this paper, the authors show that \({\mathcal C}^{\Gamma}\) can be resolved by a single toric modification of its ambient space \(\mathbb C ^{g+1}\) (\(g\) is the number of Puiseux exponents of \({\mathcal C}\)). Furthermore, \({\mathcal C}\) may be naturally embedded in \(\mathbb C ^{g+1}\) in such a way that the toric modification above resolves both \({\mathcal C}\) and \({\mathcal C}^{\Gamma}\).
This result leads to a difficult question: Given any germ of an analytic space and a valuation of its local algebra, can the germ be embedded in an affine space in such a way that a toric modification of the ambient space can resolve the singularity at the center of the valuation?
In his thesis (2002), Pedro Gonzalez Perez proved that, in the same way, an irreducible quasi-ordinary hypersurface singularity can be resolved with one toric modification of a suitable ambient space where the quasi-ordinary singularity is a deformation of a toric singularity. In the general case, the graded ring gr\(_{\nu}(R)\) is not always Noetherian and the structure of the semigroup \(\Gamma\) can be very complicated. So the authors’ question is far from being easy.
For the entire collection see [Zbl 0932.00042].

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
32S45 Modifications; resolution of singularities (complex-analytic aspects)