2015 | OriginalPaper | Buchkapitel
The smooth loci of spiral Schubert varieties of type
verfasst von : William Graham, Wenjing Li
Erschienen in: Representations of Reductive Groups
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Spiral Schubert varieties are conjecturally the only Schubert varieties in type Ã2 $$\widetilde{A}_{2}$$ for which rational smoothness at a torus-fixed point is not detected by the number of torus-invariant curves passing through that point. In this paper we determine the locus of smooth points of a spiral Schubert variety of type Ã2 $$\widetilde{A}_{2}$$ . This continues the study begun in [7], where the locus of rationally smooth points was determined. The main result describes the smooth locus in terms of the action of the Weyl group on ℝ2 $$\mathbb{R}^{2}$$ ; using this result, we identify the maximal singular points of these varieties. We make key use of the results of [7] relating the Bruhat order to the Weyl group action on ℝ2 $$\mathbb{R}^{2}$$ .