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Normality of Schubert varieties. (English) Zbl 0651.14030
Let G be a reductive algebraic group defined over an algebraically closed field. Moreover let Q be a parabolic subgroup. At least three proofs of the normality of the Schubert subvarieties in G/Q are known [H. H. Andersen, Invent. Math. 79, 611-618 (1985; Zbl 0591.14036), S. Ramanan and A. Ramanathan, Invent. Math. 79, 217-224 (1985; Zbl 0553.14023), C. S. Seshadri, Proc. Bombay Colloquium on vector bundles 1984)]. In the present article it is shown that the normality is an easy consequence of the fact that the Schubert varieties are Frobenius split proved by V. B. Mehta and A. Ramanathan in Ann. Math., II. Ser. 122, 27-40 (1986; Zbl 0601.14043)] and the following lemma:
Let $$f: Y\to X$$ be a proper surjective morphism of irreductible varieties in characteristic p. Suppose that Y is normal, the fibres of f are connected and that X is Frobenius split. Then X is normal.
Reviewer: D.Laksov

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14M17 Homogeneous spaces and generalizations
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