ADE-bundle over rational surfaces, configuration of lines and rulings
نویسنده
چکیده
To each del Pezzo surface Xn (resp. ruled surface, ruled surface with a section), we describe a natural Lie algebra bundle of type En (resp. Dn−1, An−2) over it . Using lines and rulings on any such surface, we describe various representation bundles corresponding to fundamental representations of the corresponding Lie algebra. When we specify a geometric structure on the surface to reduce the Lie algebra to a smaller one, then the classical geometry of the configuration of lines and rulings is encoded beautifully by the branching rules in Lie theory. We discuss this relationship in details. When we degenerate the surface to a non-normal del Pezzo surface, we discover that the configurations of lines and rulings are also governed by certain branching rules. However the degeneration theory of the bundle is not fully understood yet. ∗This paper is partially by a NSF grant, DMS-9803616.
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