Strickland, E. Projective spaces in flag varieties. (English) Zbl 0926.14030 Commun. Algebra 26, No. 5, 1651-1655 (1998). 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 Keywords:flag varieties; Borel subgroups; Schubert cycles PDFBibTeX XMLCite \textit{E. Strickland}, Commun. Algebra 26, No. 5, 1651--1655 (1998; Zbl 0926.14030) Full Text: DOI Link References: [1] Fulton W., Intersection Theory (1984) · Zbl 0541.14005 [2] Hartshorne R., Algebraic Geometry (1977) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.