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On Branching Indices of Affine A-D-E Diagrams : A Geometrical Characterization by Kleinian Singularities

Shi-shyr Roan
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Abstract

The exceptional configuration of the minimal resolution $\hat{S}_G $ of a Kleinian quotient surface $S_G (:= \CZ^2/G)$ 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 $S_G$ as the multiplicities of its divisor components in $\hat{S}_G$, 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 $S_G$ near the singular point whose divisors in $\hat{S}_G $ display the affine $A$-$D$-$E$ diagram configuration.

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Shi-shyr Roan. 2004-10-07. On Branching Indices of Affine A-D-E Diagrams : A Geometrical Characterization by Kleinian Singularities. https://arxiv.org/abs/math/0405580

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