Some real and unreal enumerative geometry for flag manifolds

Michigan Mathematical Journal

Volumn 48, Issue SUPPL., 2000, Pages 573-592

  Sottile, Frank ^a  

^a UNIVERSITY OF MASSACHUSETTS   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0002321615     PISSN: 00262285     EISSN: None     Source Type: Journal    
DOI: 10.1307/mmj/1030132734     Document Type: Article

References (40)

1. N. Bergeron and F. Sottile, Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J. 95 (1998), 373-423.
2. _, A Pieri-type formula for isotropic flag manifolds, www.arXiv.org/ math.CO/9810025.
3. A. Borel, Linear algebraic groups, Springer-Verlag, Berlin, 1991.
4. C. Chevalley, Sur les décompositions cellulaires des espaces G/B, Algebraic groups and their generalizations: Classical methods, pp. 1-23, Amer. Math. Soc., Providence, RI, 1994.
5. P. Dietmaier, The Stewart-Gough platform of general geometry can have 40 real postures, Advances in robot kinematics: Analysis and control, pp. 7-16, Kluwer, Dordrecht, 1998.
6. D. Eisenbud and J. Harris, Divisors on general curves and cuspidal rational curves, Invent. Math. 74 (1983), 371-418.
7. _, When ramification points meet, Invent. Math. 87 (1987), 485-493.
8. A. Eremenko and A. Gabrielov, Rational functions with real critical points and B. and M. Shapiro conjecture in real enumerative geometry, unpublished manuscript, 1999.
9. J.-C. Faugère, F. Rouillier, and P. Zimmermann, private communication, 1999.
10. W. Fulton, Intersection theory, Ergeb. Math. Grenzgeb. (3), 2, Springer-Verlag, Berlin, 1984.
11. _, Introduction to toric varieties, Ann. of Math. Stud., 131, Princeton Univ. Press, Princeton, NJ, 1993.
12. _, Introduction to intersection theory in algebraic geometry, CBMS Regional Conf. Ser. in Math., 54, Amer. Math. Soc., Providence, RI, 1996.
13. _, Young tableaux: With applications to representation theory and geometry, London Math. Soc. Stud. Texts, 35, Cambridge Univ. Press, Cambridge, U.K., 1997.
14. W. Fulton and P. Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Math., 1689, Springer-Verlag, Berlin, 1998.
15. G.-M. Greuel, G. Pfister, and H. Schönemann, Singular version 1.2 user manual, technical report no. 21, Centre for Computer Algebra, University of Kaisers-lautern, 1998.
16. B. Huber, F. Sottile, and B. Sturmfels, Numerical Schubert calculus. Symbolic numeric algebra for polynomials, J. Symbolic Comput. 26 (1998), 767-788.
17. B. Huber and J. Verschelde, Pieri homotopies for problems in enumerative geometry applied to pole placement in linear systems control, SIAM J. Control Optim. 38 (2000), 1265-1287.
18. I. Itenberg and M.-F. Roy, Multivariate Descartes' rule, Beiträge Algebra Geom. 37 (1996), 337-346.
19. V. Kharlamov, personal communication, 1999.
20. S. L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287-297.
21. F. Knop, The Luna-Vust theory of spherical embeddings, Proc. Hyderabad Conf. Alg. Groups, pp. 22-249, Manoj Prakashan, Madras, 1991.
22. A. Lascoux and M.-P. Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 447-450.
23. T. Y. Li and X. Wang, On multivariate Descartes' rule - a counterexample, Beiträge Algebra Geom. 39 (1998), 1-5.
24. 3, Discrete Comput. Geom. (to appear).
25. J. D. Monk, The geometry of flag manifolds, Proc. London Math. Soc. (3) 9 (1959), 253-286.
26. P. Pedersen and B. Sturmfels, Mixed monomial bases, Progr. Math., 143, pp. 307-316, Birkhäuser, Basel, 1994.
27. P. Pragacz and J. Ratajski, A Pieri-type formula for Lagrangian and odd orthogonal Grassmannians, J. Reine Agnew. Math. 476 (1996), 143-189.
28. _, A Pieri-type theorem for even orthogonal Grassmannians, preprint, Max-Planck Institut, 1996.
29. M. S. Ravi, J. Rosenthal, and X. Wang, Degree of the generalized Plücker embedding of a Quot scheme and quantum cohomology, Math. Ann. 311 (1998), 11-26.
30. F. Ronga, A. Tognoli, and Th. Vust, The number of conics tangent to five given conies: The real case, Rev. Mat. Univ. Complut. Madrid 10 (1997), 391-421.
31. F. Sottile, Pieri's formula for flag manifolds and Schubert polynomials, Ann. Inst. Fourier (Grenoble) 46 (1996), 89-110.
32. _, Enumerative geometry for the real Grassannian of lines in protective space, Duke Math. J. 87 (1997), 59-85.
33. _, Real enumerative geometry and effective algebraic equivalence, J. Pure Appl. Algebra 117/118 (1997), 601-615.
34. _, The special Schubert calculus is real, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 35-39.
35. _, Real Schubert calculus: Polynomial systems and a conjecture of Shapiro and Shapiro, Exper. Math. 9 (2000), 161-182.
36. _, Real rational curves in Grassmannians, J. Amer. Math. Soc. 13 (2000), 333-341.
37. F. Sottile and B. Sturmfels, A sagbi basis for the quantum Grassmannian, J. Pure Appl. Alg. (to appear).
38. B. Sturmfels, On the number of real roots of a sparse polynomial system, Fields Inst. Commun., 3, pp. 137-143, Amer. Math. Soc., Providence, RI, 1994.
39. _, Polynomial equations and convex polytopes, Amer. Math. Monthly 105 (1998) 907-922.
40. J. Verschelde, Numerical evidence of a conjecture in real algebraic geometry, Experiment. Math. 9 (2000), 183-196.