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Two aspects of log terminal surface singularities. (English) Zbl 0874.14020
Log canonical singularities of a surface $$X$$ have an index, i.e. a smallest positive integer $$I$$ such that $$I\cdot K_X$$ is Cartier. If $$\sigma:\widetilde{X}\to X$$ is a good resolution of $$X$$, $$E_i$$ being the exceptional curves, $$I$$ is obtained from the codiscrepancy $$\Delta$$, where $$\sigma^*K_X= K_{\widetilde{X}}+ \Delta$$ and $$\Delta=\sum \delta_iE_i$$ and $$I=\min\{n\in \mathbb{N}_+\mid \forall i:n\cdot\delta_i\in \mathbb{Z}\}$$. This paper studies log canonical and log terminal singularities, i.e. such singularities with $$\delta_i\leq 1$$ for all $$i$$ ($$\delta_i<1$$, respectively), for a fixed number $$I$$. There is a classification given in theorem A: The set of log terminal singularities consists of finitely many series (the number of series is explicitly given for each $$I$$) and in each series the singularities are indexed by natural numbers and “behave very regular”. Essential for this investigation is a new numerical characterization of cyclic quotient singularities: The Hirzebruch-Jung continued fraction and the description of Riemenschneider, respectively, are compared with a triplet $$(I,m,\alpha)$$ involving the index $$I$$, where $$n=I\cdot m$$, $$q=m\cdot\alpha-1$$ if the singularity is of type $$(n,q)$$. A series of cyclic quotient singularities is obtained simply in the form $$(I,m,\alpha), (I,m+I,\alpha), (I,m+2I,\alpha),\dots$$ – The theorem provides an algorithm to obtain all log canonical singularities of given index $$I$$. The author points out that his recent implementation on a computer proved useful to find log surfaces of prescribed index and Kodaira dimension 0.
In theorem B, the weights $$b_i:= E_i^2$$ of the singularity are considered. They are shown to measure “how singular” is the singularity for the case of log terminal singularities, whereas this is not the case in general. – As an application, a new bound for the maximal number of weights $$\neq 2$$ of a log terminal singularity of index $$I$$ is given. Further, ascending and descending chain conditions for maximum and minimum, respectively of the $$\delta_i$$ are considered.
The author points out a partial overlap with a paper by A. Alexeev [Duke Math. J. 69, No. 3, 527-545 (1993; Zbl 0791.14006)].
Reviewer: M.Roczen (Berlin)

##### MSC:
 14J17 Singularities of surfaces or higher-dimensional varieties 14C20 Divisors, linear systems, invertible sheaves 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14B05 Singularities in algebraic geometry
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##### References:
 [1] V. Alexeev, Fractional Indices of Log Del Pezzo Surfaces. Math. USSR Izv.33, No. 3 (1989), 613–629. · Zbl 0724.14023 [2] V. Alexeev, Two Two-dimensional Terminations. Duke Math J.69, No. 3 (1993), 527–545. · Zbl 0791.14006 [3] V.I. Arnold, Critical Points of Smooth Functions and their Normal Forms. Russ. Math. Surv.30, No. 5 (1975), 1–75. · Zbl 0343.58001 [4] R. Blache, Moishezon-Flächen mit log-canonischen Singularitäten. Ruhr-Universität Bochum, Dissertation 1992. · Zbl 0826.14002 [5] R. Blache, The Structure of l.c. Surfaces of Kodaira dimension Zero, I. Ruhr-Universität Bochum, preprint 1992, to appear in J. of Algebraic Geometry. [6] E. Brieskorn, Rationale Singularitäten komplexer Flächen. Invent. Math.4 (1968), 336–358. · Zbl 0219.14003 [7] F. Hirzebruch, Über vierdimensionale Riemannsche Flächen mehrdeutigeranalytischer Funktionen von zwei Veränderlichen. Math. Ann.126 (1953), 1–22. · Zbl 0093.27605 [8] Y. Kawamata, Crepant Blowing-up of 3-dimensional Canonical Singularities and its Application to Degenerations of Surfaces. Ann. of Math.127 (1988), 93–163. · Zbl 0651.14005 [9] F.W. Knöller, Zweidimensionale Singularitäten und Differentialformen. Math. Ann.206 (1973), 205–213. · Zbl 0258.32002 [10] J. Kollár et al, Flips and Abundance for Algebraic Threefolds. Astérisque211 (1992). [11] D. Mumford, The Topology of Normal Singularities of an Algebraic Surface and a Criterion for Simplicity. Publ. Math. I.H.E.S.9 (1961), 5–22. · Zbl 0108.16801 [12] V.V. Nikulin, Del Pezzo Surfaces with Log Terminal Singularities, I. Math. USSR Sbornik66 (1990), 231–248. · Zbl 0704.14030 [13] M. Reid, Canonical 3-Folds. In: Proceed. Géométrie Algébrique d’Angers 1979, A. Beauville ed., Sijthoff & Noordhoff 1980, 273–310. [14] O. Riemenschneider, Deformationen von Quotientensingularitäten (nach zykli-schen Gruppen). Math. Ann.209 (1974), 211–248. · Zbl 0275.32010 [15] F. Sakai, Classification of Normal Surfaces, Bowdoin 1985, Proceed, of Symp. in Pure Math.46 (1987), 451–165. [16] D. Van Straten, Weakly Normal Surface Singularities and their Improvements, Leiden, Thesis 1987. · Zbl 0638.14001
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