On Branching Indices of Affine A-D-E Diagrams : A Geometrical Characterization by Kleinian Singularities
Abstract
The exceptional configuration of the minimal resolution SG of a Kleinian quotient surface SG (:= 2/G) is depicted by a A-D-E Coxeter-Dynkin diagram. In this article, we show that branching indices of the affine A-D-E diagram is geometrically characterized by a certain special function F of SG as the multiplicities of its divisor components in SG, a version parallel to the elliptic fibration near certain types of simple singular fibers in Kodaira's elliptic surface theory. We further obtain the uniqueness property of the function F (modular local units) among all local functions in SG near the singular point whose divisors in SG display the affine A-D-E diagram configuration.
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