A geometric definition of Gabrielov numbers

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• 作者：Wolfgang Ebeling (1)
  Atsushi Takahashi (2)
  
• 关键词：Cusp singularity ; Group action ; Crepant resolution ; McKay correspondence ; Coxeter–Dynkin diagram ; Gabrielov numbers ; 32S25 ; 32S55 ; 14E16 ; 14L30
• 刊名：Revista Matem篓垄tica Complutense
• 出版年：2014
• 出版时间：July 2014
• 年：2014
• 卷：27
• 期：2
• 页码：447-460
• 全文大小：
• 参考文献：1. Arnold, V. I.: Critical points of smooth functions and their normal forms. Usp. Math. Nauk. 30(5), 3-5 (1975) (Engl. translation in, Russ. Math. Surv. 30(5), 1-5 (1975))
  2. Dolgachev, I.V.: Quotient-conical singularities on complex surfaces. Funkcional. Anal. i Prilo?en. 8(2), 75-6 (1974) (Engl. translation in, Funct. Anal. Appl. 8, 160-61 (1974))
  3. Ebeling, W., Takahashi, A.: Strange duality of weighted homogeneous polynomials. Compositio Math. 147, 1413-433 (2011) CrossRef
  4. Ebeling, W., Takahashi, A.: Mirror symmetry between orbifold curves and cusp singularities with group action. Int. Math. Res. Not. 2013, 2240-270 (2013)
  5. Gabriélov, A.M.: Dynkin diagrams for unimodal singularities. Funkcional. Anal. i Prilo?en. 8(3), 1- (1974) (English translation in, Funct. Anal. Appl. 8(3), 192-96 (1974))
  6. Ito, Y., Reid, M.: The McKay correspondence for finite subgroups of SL(3, \({\mathbb{C}})\) . In: Higher-dimensional complexvarieties Trento 1994, pp. 221-40. de Gruyter, Berlin (1996)
  
• 作者单位：Wolfgang Ebeling (1)
  Atsushi Takahashi (2)
  
  1. Institut für Algebraische Geometrie, Leibniz Universit?t Hannover, Postfach 6009, 30060?, Hannover, Germany
  2. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka?, 560-0043, Japan
  
• ISSN：1988-2807

文摘

Gabrielov numbers describe certain Coxeter–Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold’s strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an action of a finite abelian subgroup \(G\) of \(\mathrm{SL}(3,{\mathbb {C}})\) using the Gabrielov numbers of the cusp singularity and data of the group \(G\) . Here we consider a crepant resolution \(Y \rightarrow {\mathbb {C}}^3/G\) and the preimage \(Z\) of the image of the Milnor fibre of the cusp singularity under the natural projection \({\mathbb {C}}^3 \rightarrow {\mathbb {C}}^3/G\) . Using the McKay correspondence, we compute the homology of the pair \((Y,Z)\) . We construct a basis of the relative homology group \(H_3(Y,Z;{\mathbb {Q}})\) with a Coxeter–Dynkin diagram where one can read off the Gabrielov numbers.