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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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