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Excellent surfaces and their taut resolution. (English) Zbl 0979.14007

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, 341-373 (2000).
In this paper the author gives a proof of resolution of singularities in the case of excellent two-dimensional schemes embedded in three-space over an algebraically closed field of arbitrary characteristic. The centers of blowup are chosen inside the equimultiple locus, but might be non-reduced. The particular choice of the center and the resulting blow-up is called taut. The author follows Zariski in showing that one can reduce the situation to the case that the equimultiple locus is a normal crossing, and proves that this situation persists under taut blow-up. To prove that the algorithm terminates, he introduces an invariant \(i_a\) for every closed point \(a\). This invariant is an element of a well ordered subset of \({\mathbb Q}^4\). The first component is the multiplicity, the other three are orders of certain coefficient ideals associated to a defining local equation of the surface. These can be expressed with the help of the Newton polyhedron.
For the entire collection see [Zbl 0932.00042].

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
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