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Projective spaces in flag varieties. (English) Zbl 0926.14030

Introduction: Let \(G\) be a semisimple algebraic group over an algebraically closed field \(k\) of characteristic zero. Let \({\mathcal B}\) be the variety of Borel subgroups. Consider the projective embedding \({\mathcal B}\to \mathbb{P}(H^0 ({\mathcal B}, L_\rho)^*)\), where \(L_\rho\) is the line bundle associated to the Steinberg weight \(\rho\), which is the half sum of positive roots. In this paper we show that any positive dimensional projective space contained in the image of \({\mathcal B}\) is necessarily of dimension one.
Furthermore we determine exactly these lines. Indeed we show that if \(\ell\subset {\mathcal B}\) is such a line, then there exists a minimal parabolic subgroup \(P\subset G\) such that \(\ell\) is the set of Borel subgroups which are contained in \(P\). In particular this implies that the possible homology classes of such lines correspond under the usual identification of the root lattice with \(H_2({\mathcal B}, \mathbb{Z})\) to the set of simple roots.
The result is obtained as an application of some properties of the intersection of Schubert cycles in the cohomology ring of \({\mathcal B}\).

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14N05 Projective techniques in algebraic geometry
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References:

[1] Fulton W., Intersection Theory (1984) · Zbl 0541.14005
[2] Hartshorne R., Algebraic Geometry (1977)
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