Real rational curves in Grassmannians
Author:
Frank Sottile
Journal:
J. Amer. Math. Soc. 13 (2000), 333341
MSC (2000):
Primary 14M15, 14N35, 14P99, 65H20, 93B55
Published electronically:
October 25, 1999
MathSciNet review:
1706484
Fulltext PDF Free Access
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 planes incident on some given general subspaces, there are general real subspaces such that each of the (finitely many) incident 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.
 1.
Aaron
Bertram, Quantum Schubert calculus, Adv. Math.
128 (1997), no. 2, 289–305. MR 1454400
(98j:14067), http://dx.doi.org/10.1006/aima.1997.1627
 2.
Aaron
Bertram, Georgios
Daskalopoulos, and Richard
Wentworth, Gromov invariants for holomorphic maps
from Riemann surfaces to Grassmannians, J.
Amer. Math. Soc. 9 (1996), no. 2, 529–571. MR 1320154
(96f:14066), http://dx.doi.org/10.1090/S0894034796001907
 3.
C.
I. Byrnes, Pole assignment by output feedback, Three decades
of mathematical system theory, Lecture Notes in Control and Inform. Sci.,
vol. 135, Springer, Berlin, 1989, pp. 31–78. MR 1025786
(90k:93001), http://dx.doi.org/10.1007/BFb0008458
 4.
J. M. Clark, The consistent selection of local coordinates in linear system identification, Proc. Joint Automatic Control Conference, 1976, pp. 576580.
 5.
William
Fulton, Introduction to intersection theory in algebraic
geometry, CBMS Regional Conference Series in Mathematics,
vol. 54, Published for the Conference Board of the Mathematical
Sciences, Washington, DC; by the American Mathematical Society, Providence,
RI, 1984. MR
735435 (85j:14008)
 6.
B. Huber and J. Verschelde, Pieri homotopies for problems in enumerative geometry applied to pole placement in linear systems control, MSRI 1999028, 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, 767788. CMP 99:06
 8.
Kenneth
Intriligator, Fusion residues, Modern Phys. Lett. A
6 (1991), no. 38, 3543–3556. MR 1138873
(92k:81180), http://dx.doi.org/10.1142/S0217732391004097
 9.
Steven
L. Kleiman, The transversality of a general translate,
Compositio Math. 28 (1974), 287–297. MR 0360616
(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), no. 3, 329–352. MR 1273616
(95b:93043), http://dx.doi.org/10.1007/BF00998684
 11.
M.
S. Ravi, Joachim
Rosenthal, and Xiaochang
Wang, Dynamic pole assignment and Schubert calculus, SIAM J.
Control Optim. 34 (1996), no. 3, 813–832. MR 1384954
(97c:93051), http://dx.doi.org/10.1137/S036301299325270X
 12.
, Degree of the generalized Plücker embedding of a quot scheme and quantum cohomology, Math. Ann. 311 (1998), no. 1, 1126. CMP 98:13
 13.
Felice
Ronga, Alberto
Tognoli, and Thierry
Vust, The number of conics tangent to five given conics: the real
case, Rev. Mat. Univ. Complut. Madrid 10 (1997),
no. 2, 391–421. MR 1605670
(99d:14059)
 14.
Joachim
Rosenthal, On dynamic feedback compensation and compactification of
systems, SIAM J. Control Optim. 32 (1994),
no. 1, 279–296. MR 1255971
(95b:93084), http://dx.doi.org/10.1137/S036301299122133X
 15.
F. Sottile, Pieri's formula via explicit rational equivalence, Can. J. Math. 49 (1997), no. 6, 12811298. 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), 3539. CMP 99:11
 18.
F. Sottile and B. Sturmfels, A sagbi basis for the quantum Grassmannian, 16 pages, 1999. math. AG/9908016
 19.
Stein
Arild Strømme, On parametrized rational curves in Grassmann
varieties, Space curves (Rocca di Papa, 1985) Lecture Notes in
Math., vol. 1266, Springer, Berlin, 1987, pp. 251–272. MR 908717
(88i:14020), http://dx.doi.org/10.1007/BFb0078187
 20.
Cumrun
Vafa, Topological mirrors and quantum rings, Essays on mirror
manifolds, Int. Press, Hong Kong, 1992, pp. 96–119. MR 1191421
(94c:81193)
 1.
 A. Bertram, Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289305. 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, 529571. 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, SpringerVerlag, Berlin, 1989, pp. 3178. MR 90k:93001
 4.
 J. M. Clark, The consistent selection of local coordinates in linear system identification, Proc. Joint Automatic Control Conference, 1976, pp. 576580.
 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 1999028, 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, 767788. CMP 99:06
 8.
 K. Intriligator, Fusion residues, Mod. Phys. Lett. A 6 (1991), 35433556. MR 92k:81180
 9.
 S. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287297. 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), 329352. 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), 813832. MR 97c:93051
 12.
 , Degree of the generalized Plücker embedding of a quot scheme and quantum cohomology, Math. Ann. 311 (1998), no. 1, 1126. 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), 391421. MR 99d:14059
 14.
 J. Rosenthal, On dynamic feedback compensation and compactification of systems, SIAM J. Control Optim. 32 (1994), 279296. MR 95b:93084
 15.
 F. Sottile, Pieri's formula via explicit rational equivalence, Can. J. Math. 49 (1997), no. 6, 12811298. 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), 3539. 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, SpringerVerlag, 1987, pp. 251272. MR 88i:14020
 20.
 C. Vafa, Topological mirrors and quantum rings, Essays on Mirror Manifolds, International Press, 1992, ed. by S.T. Yau, pp. 96119. 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 537061388
Address at time of publication:
Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 010034515
Email:
sottile@math.umass.edu
DOI:
http://dx.doi.org/10.1090/S0894034799003239
PII:
S 08940347(99)003239
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 DMS9701755.
Article copyright:
© Copyright 2000
American Mathematical Society
