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Multiplicity-free subvarieties of flag varieties. (English) Zbl 1052.14055
Avramov, Luchezar L. (ed.) et al., Commutative algebra. Interactions with algebraic geometry. Proceedings of the international conference, Grenoble, France, July 9–13, 2001 and the special session at the joint international meeting of the American Mathematical Society and the Société Mathématique de France, Lyon, France, July 17–20, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3233-6/pbk). Contemp. Math. 331, 13-23 (2003).
Let \(G\) be a complex semisimple, simply-connected algebraic group and let \(P\) be a parabolic subgroup. Consider the flag variety \(X:= G/P\) and the Schubert subvarieties \(X_w:= \overline{BwP/P}\subset G/P\) for any \(w\in W/W_P\), where \(W\) is the Weyl group of \(G\), \(W_P\) is the Weyl group of \(P\) and \(B\) is a Borel subgroup of \(G\) contained in \(P\). Then, any subvariety \(V\) of \(X\) is rationally equivalent to a linear combination of Schubert cycles \([X_w]\) with uniquely determined nonnegative integral coefficients. Then, Brion calls \(V\) multiplicity free if these coefficients are 0 or 1. Examples of multiplicity free \(V\) include the Schubert varieties \(X_w\) themselves, \(G\)-stable (irreducible) subvarieties of \(X\times X\) (under the diagonal action of \(G\)), irreducible hyperplane sections of \(X\) in its smallest projective embedding and the irreducible hyperplane sections of Schubert varieties in Grassmannians embedded by the Plücker embedding.
The main theorem of the paper under review asserts that any multiplicity-free subvariety \(V\subset X\) is normal and Cohen-Macaulay. Further, \(V\) admits a flag degeneration inside \(X\) to a reduced Cohen-Macaulay union of Schubert varieties. Hence, for any globally generated line bundle \(G\) on \(X\), the restriction map \(H^0(X,{\mathcal L})\to H^0(V,{\mathcal L})\) is surjective and \(H^i(V,{\mathcal L})= 0\) for all \(i\geq 1\). If \(L\) is ample, then \(H^i(V, {\mathcal L}^{-1})= 0\) for any \(i<\dim V\). Thus, \(V\) is arithmetically normal and Cohen-Macaulay in the projective embedding given by any ample \({\mathcal L}\).
For the entire collection see [Zbl 1020.00012].

14M15 Grassmannians, Schubert varieties, flag manifolds
14L30 Group actions on varieties or schemes (quotients)
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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