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The minimal degeneration singularities in the affine Grassmannians. (English) Zbl 1078.14016
Sei \(G\) eine einfache endlich dimensionale algebraische Gruppe über einem algebraisch abgeschlossenen Körper \(k\) der Charakteristik \(0\) und \(\mathcal G\) das affine Grassmann-Schema von \(G\). Hauptergebnis der vorliegenden Arbeit ist
Theorem A: Falls \(G\) vom einfach-geschnürten Typ ist, sind alle Singularitäten minimaler Degeneration von \(G[[z]]\)-Bahnen in \(\mathcal G\) glatt äquivalent entweder zu Kleinschen Singularitäten vom Typ A oder zu minimalen Singularitäten von Typen, die zu zusammenhängenden Dynkin-Unterdiagrammen des Dynkindiagramms von \(G\) gehören.
Für nicht einfach-geschnürte Gruppen kommen noch sog. quasi-minimale Singularitäten hinzu, die mit Methoden der Schnitt-Kohomologie und äquivarianter Multiplizitäten untersucht werden.

MSC:
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14L15 Group schemes
14M15 Grassmannians, Schubert varieties, flag manifolds
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