Dynkin Graphs and Quadrilateral Singularities

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.