A triple intersection theorem for the varieties SO(n)/Pd

S. Sertöz

Fundamenta Mathematicae (1993)

  • Volume: 142, Issue: 3, page 201-220
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Sertöz, S.. "A triple intersection theorem for the varieties SO(n)/Pd." Fundamenta Mathematicae 142.3 (1993): 201-220. <http://eudml.org/doc/211982>.

@article{Sertöz1993,
abstract = {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.},
author = {Sertöz, S.},
journal = {Fundamenta Mathematicae},
keywords = {subspaces of a quadric; Schubert calculus; intersection theory; intersection of Schubert cells},
language = {eng},
number = {3},
pages = {201-220},
title = {A triple intersection theorem for the varieties SO(n)/Pd},
url = {http://eudml.org/doc/211982},
volume = {142},
year = {1993},
}

TY - JOUR
AU - Sertöz, S.
TI - A triple intersection theorem for the varieties SO(n)/Pd
JO - Fundamenta Mathematicae
PY - 1993
VL - 142
IS - 3
SP - 201
EP - 220
AB - 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.
LA - eng
KW - subspaces of a quadric; Schubert calculus; intersection theory; intersection of Schubert cells
UR - http://eudml.org/doc/211982
ER -

References

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  1. [1] E. Artin, Geometric Algebra, Interscience, New York 1988 (c1957). 
  2. [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. 
  3. [3] İ. Dibağ, Topology of the complex varieties A s ( n ) , J. Differential Geom. 11 (1976), 499-520. 
  4. [4] W. Fulton, Intersection Theory, Springer, 1984. 
  5. [5] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York 1978. Zbl0408.14001
  6. [6] H. Hiller and B. Boe, Pieri formulas for S O 2 n + 1 / U n and S p n / U n , Adv. in Math. 62 (1986), 49-67. 
  7. [7] W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Vol. II, Cambridge University Press, 1968. Zbl0157.27501
  8. [8] S. Kleiman and D. Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061-1082. 
  9. [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. 
  10. [10] P. Pragacz, Geometric applications of symmetric polynomials, preprint, Max-Planck Institut für Mathematik, Bonn 1992. 
  11. [11] P. Pragacz and J. Ratajski, Pieri for isotropic Grassmannians: the operator approach, preprint, Max-Planck Institut für Mathematik, Bonn 1992. 

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