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Multi-cones over Schubert varieties. (English) Zbl 0615.14028
The authors prove the Cohen-Macaulayness of multicones over Schubert varieties.
Let G be a reductive algebraic group, B a Borel subgroup and X a Schubert variety in G/B. Let $$L_ i$$, $$1\leq i\leq n$$ be the line bundles corresponding to the fundamental weights (here, $$n=rank(G))$$. Let $$L=\otimes^{n}_{i=1}L_ i^{a_ i},\quad a_ i\in {\mathbb{Z}}^+$$, $$1\leq i\leq n$$. Let $$R=\oplus_{L}H^ 0(X,L)\quad$$ and $$C=Spec(R)$$. The authors prove that the ring C is Cohen-Macaulay.
In the course of proving this, they also give a criterion for C to have rational singularities, if X does (similar to Serre’s criterion for arithmetic Cohen-Macaulayness). The proof of the main theorem uses Frobenius splitting of Schubert varieties. The Frobenius splitting of Schubert varieties was first proved by Mehta and Ramanathan. Subsequently, Ramanathan has proved geometric properties for Schubert varieties like arithmetic normality, arithmetic Cohen-Macaulayness etc., using Frobenius splitting. The result about the Cohen-Macaulayness of multicones completes the picture regarding Cohen-Macaulay properties that arise in the context of Schubert varieties. The paper is a nice contribution to the geometric study of Schubert varieties.
Reviewer: V.Lakshmibai

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds
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##### References:
 [1] Kempf, G., Knutson, F., Mumford, D., St. Donat, B.: Toroidal Embeddings I. Lecture Notes in Mathematics, vol. 339. Berlin-Heidelberg-New York: Springer 1973 [2] Mehta, V.B., Ramanathan, A.: Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. Math.122, 27-40 (1985) · Zbl 0601.14043 · doi:10.2307/1971368 [3] Ramanan, S., Ramanathan, A.: Projective normality of flag varieties and Schubert varieties. Invent. math.79, 217-224 (1985) · Zbl 0553.14023 · doi:10.1007/BF01388970 [4] Ramanathan, A.: Schubert varieties are arithmetically Cohen-Macaulay. Invent. math.80, 283-294 (1985) · Zbl 0555.14021 · doi:10.1007/BF01388607 [5] Ramanathan, A.: Equations defining Schubert varieties and Frebenius splitting of diagonals. (To appear in Pub. Math. I.H.E.S.)
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