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Properties of \(-P \cdot P\) for Gorenstein surface singularities. (English) Zbl 0901.14006
Let \((X,0)\) be the germ of a normal complex analytic surface singularity and \((\widetilde X,E)\to(X,0)\) be a good resolution of \((X,0)\), \(E\) the exceptional divisor and \(K\) the canonical divisor of \(\widetilde X\). Let \(K+E=P+N\) be the Zariski decomposition of \(K+E\). The paper concerns the properties of the topological invariant \(-P\cdot P\). The main result is that if \((X,0)\) is numerically Gorenstein and not log-canonical, then \(-P\cdot P>1/42\) with equality only for the \((2,3,7)\)-triangle singularity or its equisingular deformations. The other result is that the values of \(-P\cdot P\) for all numerically Gorenstein singularities form a well-ordered set and its smallest non-zero accumulation point is \(1/6\).

MSC:
14C20 Divisors, linear systems, invertible sheaves
14J17 Singularities of surfaces or higher-dimensional varieties
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
32B10 Germs of analytic sets, local parametrization
32S45 Modifications; resolution of singularities (complex-analytic aspects)
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