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Dynkin Graphs and Quadrilateral Singularities
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The study of hypersurface quadrilateral singularities can be§reduced to the study of elliptic K3 surfaces with a singular§fiber of type I 0 (superscript , subscript 0), and§therefore these notes consider, besides the topics of the§title, such K3 surfaces too.§The combinations of rational double points that can occur on§fibers in the semi-universal deformations of quadrilateral§singularities are examined, to show that the possible§combinations can be described by a certain law from the§viewpoint of Dynkin graphs. This is equivalent to saying§that the possible combinations of singular fibers in§elliptic K3 surfaces with a singular fiber of type I 0§(superscript , subscript 0) can be described by a certain§law using classical Dynkin graphs appearing in the theory§of semi-simple Lie groups. Further, a similar description§for thecombination of singularities on plane sextic curves§is given. Standard knowledge of algebraic geometry at the§level of graduate students is expected. A new method based§on graphs will attract attention of researches.
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