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Birational transformations of weighted graphs. (English) Zbl 1129.14056

Hibi, Takayuki (ed.), Affine algebraic geometry. Dedicated to Masayoshi Miyanishi on the occasion of his retirement from Osaka University. Osaka: Osaka University Press (ISBN 978-4-87259-226-9/hbk). 107-147 (2007); corrigendum in: Affine algebraic geometry: The Russell Festschrift. Outgrow of an international conference, McGill University, Montreal, QC, Canada. June 1–5, 2009, held in honour of Professor Peter Russell on the occasion of his 70th birthday. Providence, RI: American Mathematical Society (AMS). CRM Proceedings and Lecture Notes 54, 35-38 (2011).
This paper is a part of the huge project of the authors aiming to
1) classify all smooth affine surfaces admitting a \(\mathbb C^*\)-action (\(\mathbb C^*\)-surfaces);
2) classify \(\mathbb C^*\)-actions on a specific \(\mathbb C^*\)-surface \(S\).
One of the main tools for such classification is the weighted graph of the divisor \(D=\overline S-S\) at infinity, where \(\overline S\) is an equivariant compactification of \(S.\)
The goal of this paper is classification of weighted graphs. The authors introduce the notion of a standard model of a graph and classify the standard models. The birational transformations between standard models are described and are related to the elementary transformations of ruled surfaces.
For the entire collection see [Zbl 1117.14002].

MSC:

14J50 Automorphisms of surfaces and higher-dimensional varieties
14J26 Rational and ruled surfaces
14R99 Affine geometry
05C99 Graph theory
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