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Equations defining Schubert varieties and Frobenius splitting of diagonals. (English) Zbl 0634.14035

Let G be a simply connected semisimple algebraic group over an algebraically closed field. Let B be a Borel subgroup and \(Q\supset B\) a parabolic subgroup. A generalized Schubert variety in G/Q is the closure of a B-orbit in G/Q. Musili, Laskhmibai and Seshadri have used the theory of standard monomials to prove that the homogeneous ideal of G/Q in any embedding given by a very ample line bundle is generated by quadrics and that the homogeneous ideal of any Schubert variety in the homogeneous coordinate ring of G/Q is generated by linear homogeneous polynomials vanishing on it for all the classical groups G, and they conjectured this result to be valid for general G/Q [see V. Lakshmibai and C. S. Seshadri, J. Algebra 100, 462-557 (1986; Zbl 0618.14026)]. The main result of the paper under review proves this conjecture over fields of arbitrary characteristic. To do this the author works first over fields of positive characteristic and uses the notion of Frobenius split varieties introduced by V. B. Mehta and himself [Ann. Math., II. Ser. 122, 27-40 (1985; Zbl 0601.14043)].
For a variety X over a base field k of characteristic p\(>0\) one has the absolute Frobenius morphism \(F: X\to X\) given by the ring homomorphism \(a\mapsto a^ p\). The variety X is called Frobenius split if the p-th power map \({\mathcal O}_ X\to F_*{\mathcal O}_ X\) has a section \(\phi: F_*{\mathcal O}_ X\to {\mathcal O}_ X.\) A closed subvariety \(Y\subset X\) is called compatibly Frobenius split in X if \(\phi (F_ XI)=I\), where I is the ideal sheaf of Y in X. Let \(X_ r=X\times...\times X\) \((r\quad factors)\) and \(\Delta_ r\) the diagonal in \(X_ r\). Put \(\Delta_{12}=\Delta_ 2\times X\) and \(\Delta_{23}=X\times \Delta_ 2\). The author proves that if X satisfies the conditions \((a)\quad \Delta_ 2\) in \(X_ 2\) is compatibly Frobenius split and \((b)\quad \Delta_{12}\cup \Delta_{23}\) in \(X_ 3\) is compatibly split, then in any projective embedding of X given by an ample line bundle the homogeneous ideal of X is generated by quadrics, and that a Cartier divisor Y of X is defined by linear equations if \(\Delta_ 2\cup Y\times X\) is compatibly Frobenius in \(X_ 2\). To check these conditions for Schubert varieties the author proceeds as follows.
He knows from the paper by V. B. Mehta and himself [cited above] that G/B has a Frobenius splitting \(\phi\) which compatibly splits all Schubert varieties in G/B. The product splitting \(\phi\times \phi\) on G/B\(\times G/B\) will compatibly split the factor G/B\(\times 0\) since the point Schubert variety 0 is compatibly split in G/B. So one needs only to find an automorphism of G/B\(\times G/B\) which pulled the factor into the diagonal. To do that one considers the unipotent radical \(\tilde U\) of the opposite Borel subgroup as an open subset of G/B and the automorphism \(\alpha\) of \(\tilde U\times \tilde U\) with such property given by \(\alpha (x,y)=(x,y^{-1}x)\). Then one shows that \(\alpha^{-1}(\phi \times \phi)\), which is a splitting of the diagonal in \(\tilde U\times \tilde U\), extends to the whole of G/B\(\times G/B\). Similarly, one uses an automorphism of \(\tilde U\times \tilde U\times \tilde U\) to prove that \(\Delta_{12}\cup \Delta_{23}\) is compatibly split in G/B\(\times G/B\times G/B.\)
Finally, the author uses semicontinuity to pass from characteristic p to characteristic 0. The result on the linear definition of Schubert varieties in G/Q together with the fact that the intersection of any set of Schubert varieties is reduced [see the author, Invent. Math. 80, 283- 294 (1985; Zbl 0541.14039)] can be used to study other properties of Schubert varieties. One can also apply the method of this paper to study the relations of higher syzygies of Schubert varieties.
Reviewer: Ngo Viet Trung

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14M10 Complete intersections
14G15 Finite ground fields in algebraic geometry
20G15 Linear algebraic groups over arbitrary fields
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