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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
Keywords:
Schubert variety; small resolution
Full Text:
References:
 [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. [6] Lakshmibai, V., Singular loci of Schubert varieties for classical groups, Bull. A.M.S., 16 (1987), 83-90. · Zbl 0635.14022 · doi:10.1090/S0273-0979-1987-15466-8 [7] Zelevinskii, A. V., Small resolutions of singularities of Schubert varieties, FunktsionaFnye Analiii Ego Prilozheniya, 17 (1983), 75-77.
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