Dynkin Graphs and Quadrilateral Singularities
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The study of hypersurface quadrilateral singularities can bereduced to the study of elliptic K3 surfaces with a singularfiber of type I _ 0 (superscript _, subscript 0), andtherefore these notes consider, besides the topics of thetitle, such K3 surfaces too.The combinations of rational double points that can occur onfibers in the semi-universal deformations of quadrilateralsingularities are examined, to show that the possiblecombinations can be described by a certain law from theviewpoint of Dynkin graphs. This is equivalent to sayingthat the possible combinations of singular fibers inelliptic...
The study of hypersurface quadrilateral singularities can bereduced to the study of elliptic K3 surfaces with a singularfiber of type I _ 0 (superscript _, subscript 0), andtherefore these notes consider, besides the topics of thetitle, such K3 surfaces too.The combinations of rational double points that can occur onfibers in the semi-universal deformations of quadrilateralsingularities are examined, to show that the possiblecombinations can be described by a certain law from theviewpoint of Dynkin graphs. This is equivalent to sayingthat the possible combinations of singular fibers inelliptic K3 surfaces with a singular fiber of type I _ 0(superscript _, subscript 0) can be described by a certainlaw using classical Dynkin graphs appearing in the theoryof semi-simple Lie groups. Further, a similar descriptionfor thecombination of singularities on plane sextic curvesis given. Standard knowledge of algebraic geometry atthelevel of graduate students is expected. A new method basedon graphs will attract attention of researches.
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