We study the Schubert calculus on the space of d-dimensional linear subspaces of a smooth n-dimensional quadric lying in the projective space. Following Hodge and Pedoe we develop the intersection theory of this space in a purely combinatorial manner. We prove in particular that if a triple intersection of Schubert cells on this space is nonempty then a certain combinatorial relation holds among the Schubert symbols involved, similar to the classical one. We also show when these necessary conditions are also sufficient to obtain a nontrivial intersection. Several examples are calculated to illustrate the main results.
@article{bwmeta1.element.bwnjournal-article-fmv142i3p201bwm,
author = {S. Sert\"oz},
title = {A triple intersection theorem for the varieties SO(n)/Pd},
journal = {Fundamenta Mathematicae},
volume = {142},
year = {1993},
pages = {201-220},
zbl = {0837.14040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv142i3p201bwm}
}
Sertöz, S. A triple intersection theorem for the varieties SO(n)/Pd. Fundamenta Mathematicae, Tome 142 (1993) pp. 201-220. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv142i3p201bwm/
[00000] [1] E. Artin, Geometric Algebra, Interscience, New York 1988 (c1957).
[00001] [2] I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Schubert cells and the cohomology of G/P spaces, Russian Math. Surveys 28 (1973), 1-26.
[00002] [3] İ. Dibağ, Topology of the complex varieties , J. Differential Geom. 11 (1976), 499-520.
[00003] [4] W. Fulton, Intersection Theory, Springer, 1984.
[00004] [5] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York 1978. | Zbl 0408.14001
[00005] [6] H. Hiller and B. Boe, Pieri formulas for and , Adv. in Math. 62 (1986), 49-67.
[00006] [7] W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Vol. II, Cambridge University Press, 1968. | Zbl 0157.27501
[00007] [8] S. Kleiman and D. Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061-1082.
[00008] [9] P. Pragacz, Algebro-geometric applications of Schur S- and Q-polynomials, in: Topics in Invariant Theory, Séminaire d'Algèbre Dubreil-Malliavin 1989-1990, Lecture Notes in Math. 1478, Springer, 1991, 130-191.
[00009] [10] P. Pragacz, Geometric applications of symmetric polynomials, preprint, Max-Planck Institut für Mathematik, Bonn 1992.
[00010] [11] P. Pragacz and J. Ratajski, Pieri for isotropic Grassmannians: the operator approach, preprint, Max-Planck Institut für Mathematik, Bonn 1992.