Tie transformations of Dynkin graphs and singularities on quartic surfaces
type: Item
instance of: scholarly article
title: Tie transformations of Dynkin graphs and singularities on quartic surfaces
zbMATH Open document ID: 0667.14017
DOI: 10.1007/BF01231185
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143783
review text: This article is the continuation of my previous one in Invent Math. 87, 549-572 (1987; Zbl 0612.14035). We continue to study possible combinations of rational double points on quartic surfaces in the projective space \(\mathbb{P}^3\) of dimension 3. Now in the cited paper I proposed a certain operation on Dynkin graphs, which I called an elementary transformation. This notion was natural and simple. We would like to propose another procedure, called a tie transformation. Performing such a transformation can increase the number of vertices in the Dynkin graph by one. In this, it differs from elementary transformations, since an elementary transformation never makes the number of vertices larger. Because of this property of tie transformations we can treat many interesting examples of K3 surfaces whose Picard number is maximal 20. A systematic description for possible combinations of rational double points on quartic surfaces by these two kinds of transformations of Dynkin graphs is given. Indeed, the following is a part of the main results: Theorem 0.2. Any Dynkin graph without vertices corresponding to short roots obtained from one of the following 9 basic Dynkin graphs by elementary or tie transformations repeated twice (Four kinds of combinations - i.e., ``elementary'' twice, ``tie'' twice, ``elementary'' after ``tie'', and ``tie'' after ``elementary'' - are all permitted.) describes the combination of singularities of a quartic surface in the projective space of dimension 3 with only rational double points as singularities. - The basic Dynkin graphs: \(A_{11} + E_6\), \(2D_8 + B_1\), \(D_{16} + B_ 1\), \(2E_ 8 + B_1\), \(A_{15} + B_2\), \(2E_7 + B_3\), \(D_{12} + B_5\), \(E_8 + B_9\), \(B_{17}\). Theorem 0.4'. Let \(G=\sum a_ kA_ k +\sum b_ lD_ l +\sum c_ mE_ m \) (a finite sum) be a Dynkin graph with components of type A, D or E only. Set \(r=\sum a_ kk +\sum b_{\ell}\ell +\sum c_ mm \). Then the following conditions (A) and (B) are equivalent: (A) There exists …
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14E99
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 4091649
zbMATH Keywords: rational double points on quartic surfaces
zbMATH Keywords: tie transformation
zbMATH Keywords: Dynkin graph
zbMATH Keywords: K3 surfaces whose Picard number is maximal 20
author: Tohsuke Urabe
reviewed by: Tohsuke Urabe
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4091649