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A \(p_ g\)-formula and elliptic singularities. (English) Zbl 0589.14013

The paper is concerned with normal two dimensional singularities (V,p), in particular with elliptic singularities in the sense of Wagreich \((p_ a=1)\). The goal is, to compare Zariski’s canonical resolution (successive blowing up points and normalization) with the minimal resolution of (V,p). One main result is the following:
Let (V,p) be elliptic and Gorenstein and let E be the minimal elliptic cycle on the minimal resolution of (V,p). - \((i):\quad (V,p)\) is absolutely isolated (i.e. no normalization occurs in Zariski’s canonical resolution) if and only if \(E^ 2\leq -3\). - \((ii):\quad Zariski's\) canonical resolution gives the minimal resolution if and only if \(E^ 2\leq -2\). - This extends known result of H. B. Laufer and S. S.-T. Yau. Moreover the author gives a precise description of the relation between Zariski’s canonical resolution and the minimal resolution without any assumption on \(E^ 2.\)
In the proofs the author uses a formula for the geometric genus \(p_ g\) of a normal 2-dimensional hypersurface singularity in terms of resolution data obtained by successive blowing up smooth centers. This formula is proved in the first part of the paper.
Reviewer: G.-M.Greuel

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
14B05 Singularities in algebraic geometry
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