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Small resolutions of Schubert varieties in symplectic and orthogonal Grassmannians. (English) Zbl 0859.14019
Recall [M. Goresky and R. MacPherson, Invent. Math. 72, 77-129 (1983; Zbl 0529.55007)] that a resolution \(p:\widetilde{X}\to X\) of an irreducible complex variety \(X\) is said to be small if, for each \(i>0\), one has \(\text{codim}_X \{x\in X\mid\dim p^{-1}(x)\geq i\}>2i\).
Let \(G=Sp(2n,\mathbb{C})\) or \(SO(2n,\mathbb{C})\), and let \(P=\mathbb{P}_n\). Recall [V. Lakshmibai and C. S. Seshadri, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1-54 (1978; Zbl 0447.14011)] that the Schubert varieties in \(Sp(2n,\mathbb{C})/P\) are indexed by \(\bigcup_{0\leq r\leq n} I_{n,r}\) where \(I_{n,r}=\{(\lambda_,\dots,\lambda_r)\mid 1\leq\lambda_1<\cdots<\lambda_r\leq n\}\).
Similarly, the Schubert varieties in \(SO(2n)/\mathbb{P}_n\) are labelled by the set \(\bigcup_{\substack{ 0\leq r\leq n\\ (n-r)\text{ even}}} I_{n,r}\).
The main results of this paper are:
Theorem 1.1. Let \(\lambda=(\lambda_1,\dots,\lambda_r)\in I_{n,r}\).
(i) The Schubert variety \(X(\lambda)\subset Sp(2n)/\mathbb{P}_n\) has a small resolution if \(\lambda_r\leq n-r\).
(ii) Assume \(n-r\) is even so that \(\lambda\) gives rise to a Schubert variety \(X(\lambda)\) in \(SO(2n)/\mathbb{P}_n\). \(X(\lambda)\) has a small resolution if: (a) \(\lambda_r<n-r\); or (b) for \(r\geq 2\), \(\lambda_r=n\), \(\lambda_{r-1}\leq n-r\).
Theorem 1.2. Let \(\lambda=(n)\), \(n\geq 3\), and let \(Q\) be any parabolic subgroup contained in \(\mathbb{P}_n\subset Sp(2n)\). Let \(X(\Lambda)\) be the inverse image of \(X(\lambda)\subset Sp(2n)/\mathbb{P}_n\) under the projection \(Sp(2n)/Q\to Sp(2n)/\mathbb{P}_n\). Then \(X(\Lambda)\) does not admit any small resolution.

MSC:
14M07 Low codimension problems in algebraic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
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[1] Bourbaki, N., Groupes et algebres de Lie, Ch. 4-6, Hermann, Paris 1968.
[2] Demazure, M., Desingularisation des varietes de Schubert generalises, Ann. Sci. E.N.S., 1 (1974), 53-88. · Zbl 0312.14009 · numdam:ASENS_1974_4_7_1_53_0 · eudml:81930
[3] Goresky, M. and MacPherson, R., Intersection Homology-II, Invent. Math., 71 (1983), 77-129. · Zbl 0529.55007 · doi:10.1007/BF01389130 · eudml:143014
[4] Hansen, H., On cycles in flag manifolds, Math Scand., 33 (1973), 269-274. · Zbl 0301.14019 · eudml:166330
[5] Lakshmibai, V. and Seshadri, C. S., Geometry of G/P-ll, Proc. Ind. Acad. Sci., 87 A (1978), 1-54.
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