Ganter, Frieda M. Properties of \(-P \cdot P\) for Gorenstein surface singularities. (English) Zbl 0901.14006 Math. Z. 223, No. 3, 411-419 (1996). 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 \(1/6\). Reviewer: T.Krasiński (Łódź) Cited in 3 Documents 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) Keywords:germ of a normal complex analytic surface singularity; canonical divisor; numerical Gorenstein singularities PDF BibTeX XML Cite \textit{F. M. Ganter}, Math. Z. 223, No. 3, 411--419 (1996; Zbl 0901.14006) Full Text: DOI EuDML OpenURL References: [1] Alexeev, V.A. (1989): Fractional indices of Log Del Pezzo surfaces. Math. USSR Izv.33, no. 3, 613–629 · Zbl 0724.14023 [2] Blache, R. (1994): Two aspects of log terminal surface singularities. Abh. Math. Sem. Univ. of Hamburg64, 59–87 · Zbl 0874.14020 [3] Dolgachev, I. (1980/81):Cohomologically insignificant degenerations of algebraic varieties. Composito Math.42, no. 3, 279–313 · Zbl 0466.14003 [4] Eisenbud, D., Neumann, W. (1985): Three-dimensional Link Theory and Invariants of Plane Curve Singularities. Ann. Math.110 · Zbl 0628.57002 [5] Kawamata, Y. (1988):Crepant blowing up of 3-dimensional canonical singularities and its applications to degenerations of surfaces. Ann. Math.127, 93–163 · Zbl 0651.14005 [6] Laufer, H. (1977): On minimally elliptic singularities. Amer. J. Math.99 · Zbl 0384.32003 [7] Neumann, W.: Geometry of quasi-homogeneous surface singularities. Proc. Sympos. Pure Math.40, part 2, Amer. Math. Soc. Providence RI, 245–258 [8] Sakai, F. (1984): Anticanonical models of rational surfaces. Math. Ann.269, 389–410 · Zbl 0547.14021 [9] Thurston, W. (1979): The geometry and topology of three-manifolds. The Geometry Center (unpublished) [10] Wahl, J. (1990): A characteristic number for links of surface singularities. J. Amer. Math. Soc.3, 625–637 · Zbl 0743.14026 [11] Wahl, J. (1983): Derivations, automorphisms and deformations of quasi-homogeneous singularities. Proc. Sympos. Pure Math.40, part 2 · Zbl 0534.14001 [12] Wahl, J. (1987): The jacobian algebra of a graded Gorenstein singularity. Duke Math. J.55, no. 4 · Zbl 0644.14001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.