Complex surface singularities from the combinatorial point of view
type: Item
instance of: scholarly article
title: Complex surface singularities from the combinatorial point of view
zbMATH Open document ID: 0840.14022
DOI: 10.1016/0166-8641(95)00038-I
author: F. Larrión
author: José Seade
published in: Topology and its Applications
publication date: 1996-07-04T00:00:00Z
review text: In this paper the interplay between graph theory and the topology of isolated complex surface singularities is studied. To begin with it is well known that the dual graph \((\Gamma, w,g)\) of a good resolution \(\pi: \widetilde {V}\to V\) of a normal surface singularity \((V,p)\) determines \(\widetilde {V}\) up to diffeomorphism. Here the vertices of the graph \(\Gamma\) correspond to the irreducible components \(E_i\) of the exceptional fibre of \(\pi\), the edges to the intersections of these components, the coordinates \(w_i\) of the vector \(w\) of weights to the self-intersection numbers and the coordinates \(g_i\) of the vector \(g\) to the genera of the Riemann surfaces \(E_i\). Moreover the intersection matrix \(E= (E_i, E_j)\) is always negative definite. On the other hand the intersection matrix \(E= \text{adj} (\Gamma)+ \text{diag} (w)\) of an (abstract) weighted graph \((\Gamma, w)\) is negative definite for almost all negative weights \(w_i\). Especially an (abstract) double-weighted graph \((\Gamma, w,g)\), \(w\in \mathbb{Z}^n\), \(g\in \mathbb{N}^n\), without loops is the dual graph of a normal surface singularity if and only if its intersection matrix is negative definite. Finally, the authors exploit this interplay in the case of Gorenstein singularities. A normal surface singularity is said to be numerically Gorenstein if the tangent bundle of \(V-p\) is topologically trivial over \(\mathbb{C}\) which amounts to say that the canonical class \(K\in H^2 (V, \mathbb{Q})\) is an integral linear combination of the \(E_i\) satisfying the adjunction formula \(2g-2= w+EK\). For an (abstract) weighted graph \((\Gamma,w)\) with non-singular intersection matrix \(E\) then it is shown that the set of vectors \(g\) of genera such that the adjunction formula \(2g-2= 2+EK\) has an integral solution \(K\) is infinite and parametrized by the integral solutions \(X\) of the congruence \(EX\equiv w(2)\). Finally the graph manifold \(X(\Gamma)\) obtained by plumbing …
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 32B99
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 823483
zbMATH Keywords: plumbing
zbMATH Keywords: complex surface singularities
zbMATH Keywords: weighted graph
zbMATH Keywords: Gorenstein singularities
zbMATH Keywords: normal surface singularity
zbMATH Keywords: spin-structure
zbMATH Keywords: rational double points
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 823483