Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Real rational curves in Grassmannians


Author: Frank Sottile
Journal: J. Amer. Math. Soc. 13 (2000), 333-341
MSC (2000): Primary 14M15, 14N35, 14P99, 65H20, 93B55
DOI: https://doi.org/10.1090/S0894-0347-99-00323-9
Published electronically: October 25, 1999
MathSciNet review: 1706484
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some given general figures. For the problem of plane conics tangent to five general (real) conics, the surprising answer is that all 3264 may be real. Similarly, given any problem of enumerating $p$-planes incident on some given general subspaces, there are general real subspaces such that each of the (finitely many) incident $p$-planes is real. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions real.


References [Enhancements On Off] (What's this?)

  • 1. A. Bertram, Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289-305. MR 98j:14067
  • 2. A. Bertram, G. Daskalopoulos, and R. Wentworth, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, J. Amer. Math. Soc. 9 (1996), no. 2, 529-571. MR 96f:14066
  • 3. C. I. Byrnes, Pole assignment by output feedback, Three Decades of Mathematical Systems Theory (H. Nijmeijer and J. M. Schumacher, eds.), Lecture Notes in Control and Inform. Sci., vol. 135, Springer-Verlag, Berlin, 1989, pp. 31-78. MR 90k:93001
  • 4. J. M. Clark, The consistent selection of local coordinates in linear system identification, Proc. Joint Automatic Control Conference, 1976, pp. 576-580.
  • 5. Wm. Fulton, Introduction to intersection theory in algebraic geometry, CBMS 54, AMS, 1984. MR 85j:14008
  • 6. B. Huber and J. Verschelde, Pieri homotopies for problems in enumerative geometry applied to pole placement in linear systems control, MSRI 1999-028, SIAM J. Control and Optim., to appear.
  • 7. B. Huber, F. Sottile, and B. Sturmfels, Numerical Schubert calculus, J. Symb. Comp. 26 (1998), no. 6, 767-788. CMP 99:06
  • 8. K. Intriligator, Fusion residues, Mod. Phys. Lett. A 6 (1991), 3543-3556. MR 92k:81180
  • 9. S. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287-297. MR 50:13063
  • 10. M. S. Ravi and J. Rosenthal, A smooth compactification of the space of transfer functions with fixed McMillan degree, Acta Appl. Math. 34 (1994), 329-352. MR 95b:93043
  • 11. M. S. Ravi, J. Rosenthal, and X.C. Wang, Dynamic pole assignment and Schubert calculus, SIAM J. Control and Optim. 34 (1996), 813-832. MR 97c:93051
  • 12. -, Degree of the generalized Plücker embedding of a quot scheme and quantum cohomology, Math. Ann. 311 (1998), no. 1, 11-26. CMP 98:13
  • 13. F. Ronga, A. Tognoli, and Th. Vust, The number of conics tangent to 5 given conics: the real case, Rev. Mat. Univ. Complut. Madrid 10 (1997), 391-421. MR 99d:14059
  • 14. J. Rosenthal, On dynamic feedback compensation and compactification of systems, SIAM J. Control Optim. 32 (1994), 279-296. MR 95b:93084
  • 15. F. Sottile, Pieri's formula via explicit rational equivalence, Can. J. Math. 49 (1997), no. 6, 1281-1298. CMP 98:09
  • 16. -, Real Schubert calculus: Polynomial systems and a conjecture of Shapiro and Shapiro, 28 pp., Exper. Math., to appear. 1999.
  • 17. -, The special Schubert calculus is real, ERA of the AMS 5 (1999), 35-39. CMP 99:11
  • 18. F. Sottile and B. Sturmfels, A sagbi basis for the quantum Grassmannian, 16 pages, 1999. math. AG/9908016
  • 19. S. A. Strømme, On parameterized rational curves in Grassmann varieties, Space Curves (F. Ghione, C. Peskine, and E. Sernesi, eds.), Lecture Notes in Math., vol. 1266, Springer-Verlag, 1987, pp. 251-272. MR 88i:14020
  • 20. C. Vafa, Topological mirrors and quantum rings, Essays on Mirror Manifolds, International Press, 1992, ed. by S.-T. Yau, pp. 96-119. MR 94c:81193

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 14M15, 14N35, 14P99, 65H20, 93B55

Retrieve articles in all journals with MSC (2000): 14M15, 14N35, 14P99, 65H20, 93B55


Additional Information

Frank Sottile
Affiliation: Department of Mathematics, University of Wisconsin, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
Address at time of publication: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003-4515
Email: sottile@math.umass.edu

DOI: https://doi.org/10.1090/S0894-0347-99-00323-9
Received by editor(s): April 29, 1999
Received by editor(s) in revised form: August 24, 1999
Published electronically: October 25, 1999
Additional Notes: Research at MSRI supported in part by NSF grant DMS-9701755.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society