Equisingularity classes of birational projections of normal singularities to a plane. (English) Zbl 1130.14004
A sandwiched singularity is a local ring \(\mathcal O\) which is a birational extension of a two-dimensional regular local ring \(R\). Let us consider a surface \(S\) which has a point whose local ring is a sandwich singularity. The authors study in this paper the equisingularity classes of all birational projections of \(S\) to a plane. This problem was dealt with in M. Spivakovky’s paper [Ann. Math. (2) 131, 411–491 (1990; Zbl 0719.14005)]. Spivakovsky divides this problem into two parts: discrete and continuous. In this paper the authors deal with the discrete part.
Any birational projection from a sandwiched singularity to a plane is obtained by the morphism of blowing up a complete \(\mathfrak m_{ O}\)-primary ideal \(J\) in the local ring of a regular point \(O\) on the plane. The goal of this paper is to give the equisingularity type of these ideals. More precisely: Let \(\mathcal O\) be a birational normal extension of a regular local ring \((R,\mathfrak m_O)\); the authors describe the equisingularity type of any complete \(\mathfrak m_O\)-primary ideal \(J\subset R\) such that its blow-up \(\text{Bl}_J(R)\) has some point \(Q\) whose local ring is analytically isomorphic to \(\mathcal O\). This is done by the Enriques diagram of the cluster of base points of any such ideal. Recall that an Enriques diagram is a tree together with a binary relation—proximity— representing topological equivalence classes of clusters of points in the plane.
Therefore, in section 1 the authors give an overview concerning the language of infinitely near points, sandwiched surface singularities and graphs. A good source for this is the book of E. Casas-Alvero [Singularities of Plane Curves. Cambridge University Press (2000; Zbl 0967.14018)]. In section two the authors introduce a technical device, namely the consept of contraction for a sandwiched singularity \(\mathcal O\). A contraction for a sandwiched singularity is the resolution graph \(\Gamma_{\mathcal O}\) of \(\mathcal O\), enriched by some proximity relations between their vertices. After having fixed a sandwich graph, the problem is to find the whole list of possibilities for such proximities. This is achieved in section 3. In section 4 the authors study the problem of describing the equisingularity classes of the ideals for a given sandwiched surface singularity, i.e., they describe all the possible Enriques diagram for \(\mathcal O\).
Any birational projection from a sandwiched singularity to a plane is obtained by the morphism of blowing up a complete \(\mathfrak m_{ O}\)-primary ideal \(J\) in the local ring of a regular point \(O\) on the plane. The goal of this paper is to give the equisingularity type of these ideals. More precisely: Let \(\mathcal O\) be a birational normal extension of a regular local ring \((R,\mathfrak m_O)\); the authors describe the equisingularity type of any complete \(\mathfrak m_O\)-primary ideal \(J\subset R\) such that its blow-up \(\text{Bl}_J(R)\) has some point \(Q\) whose local ring is analytically isomorphic to \(\mathcal O\). This is done by the Enriques diagram of the cluster of base points of any such ideal. Recall that an Enriques diagram is a tree together with a binary relation—proximity— representing topological equivalence classes of clusters of points in the plane.
Therefore, in section 1 the authors give an overview concerning the language of infinitely near points, sandwiched surface singularities and graphs. A good source for this is the book of E. Casas-Alvero [Singularities of Plane Curves. Cambridge University Press (2000; Zbl 0967.14018)]. In section two the authors introduce a technical device, namely the consept of contraction for a sandwiched singularity \(\mathcal O\). A contraction for a sandwiched singularity is the resolution graph \(\Gamma_{\mathcal O}\) of \(\mathcal O\), enriched by some proximity relations between their vertices. After having fixed a sandwich graph, the problem is to find the whole list of possibilities for such proximities. This is achieved in section 3. In section 4 the authors study the problem of describing the equisingularity classes of the ideals for a given sandwiched surface singularity, i.e., they describe all the possible Enriques diagram for \(\mathcal O\).
Reviewer: Karl-Heinz Kiyek (Paderborn)
MSC:
| 14B05 | Singularities in algebraic geometry |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
| 13H05 | Regular local rings |
Keywords:
normal surface singularity; sandwiched singularity; complete ideal; Enriques diagram; equisingularityReferences:
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