Real $k$-flats tangent to quadrics in $\mathbb {R}^n$
HTML articles powered by AMS MathViewer
- by Frank Sottile and Thorsten Theobald PDF
- Proc. Amer. Math. Soc. 133 (2005), 2835-2844 Request permission
Abstract:
Let $d_{k,n}$ and $\#_{k,n}$ denote the dimension and the degree of the Grassmannian $\mathbb {G}_{k,n}$, respectively. For each $1 \le k \le n-2$ there are $2^{d_{k,n}} \cdot \#_{k,n}$ (a priori complex) $k$-planes in $\mathbb {P}^n$ tangent to $d_{k,n}$ general quadratic hypersurfaces in $\mathbb {P}^n$. We show that this class of enumerative problems is fully real, i.e., for $1 \le k \le n-2$ there exists a configuration of $d_{k,n}$ real quadrics in (affine) real space $\mathbb {R}^n$ so that all the mutually tangent $k$-flats are real.References
- P. Aluffi and W. Fulton. Lines tangent to four surfaces containing a curve, in preparation.
- A. Eremenko and A. Gabrielov, Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. of Math. (2) 155 (2002), no. 1, 105–129. MR 1888795, DOI 10.2307/3062151
- 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, DOI 10.1090/cbms/054
- W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry. Vol. I, Cambridge, at the University Press; New York, The Macmillan Company, 1947. MR 0028055
- S. L. Kleiman and Dan Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061–1082. MR 323796, DOI 10.2307/2317421
- I. G. Macdonald, J. Pach, and T. Theobald, Common tangents to four unit balls in $\Bbb R^3$, Discrete Comput. Geom. 26 (2001), no. 1, 1–17. MR 1832726, DOI 10.1007/s004540010090
- H. Schubert. Anzahlbestimmungen für lineare Räume beliebiger Dimension. Acta Math. 8:97-118, 1886.
- V. Sedykh and B. Shapiro. Two conjectures on convex curves. Preprint, 2002. arXiv:math.AG/0208218.
- Frank Sottile, Enumerative geometry for the real Grassmannian of lines in projective space, Duke Math. J. 87 (1997), no. 1, 59–85. MR 1440063, DOI 10.1215/S0012-7094-97-08703-2
- Frank Sottile, The special Schubert calculus is real, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 35–39. MR 1679451, DOI 10.1090/S1079-6762-99-00058-X
- Frank Sottile, From enumerative geometry to solving systems of polynomials equations, Computations in algebraic geometry with Macaulay 2, Algorithms Comput. Math., vol. 8, Springer, Berlin, 2002, pp. 101–129. MR 1949550, DOI 10.1007/978-3-662-04851-1_{6}
- Frank Sottile and Thorsten Theobald, Lines tangent to $2n-2$ spheres in $\Bbb R^n$, Trans. Amer. Math. Soc. 354 (2002), no. 12, 4815–4829. MR 1926838, DOI 10.1090/S0002-9947-02-03014-3
- Bernd Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, vol. 97, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2002. MR 1925796, DOI 10.1090/cbms/097
- R. Vakil. Schubert Induction. Preprint, 2003. arXiv:math.AG/0302296.
Additional Information
- Frank Sottile
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- MR Author ID: 355336
- ORCID: 0000-0003-0087-7120
- Email: sottile@math.tamu.edu
- Thorsten Theobald
- Affiliation: Institut für Mathematik, MA 6-2, Technische Universität Berlin, Strasse des 17. Juni 1936, D-10623 Berlin, Germany
- MR Author ID: 618735
- ORCID: 0000-0002-5769-0917
- Email: theobald@math.tu-berlin.de
- Received by editor(s): March 11, 2004
- Received by editor(s) in revised form: May 25, 2004
- Published electronically: April 8, 2005
- Additional Notes: The research of the first author was supported by NSF CAREER grant DMS-0070494 and the Clay Mathematical Institute
- Communicated by: Michael Stillman
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 2835-2844
- MSC (2000): Primary 14N10, 51M30, 14P99, 52C45, 05A19
- DOI: https://doi.org/10.1090/S0002-9939-05-07880-9
- MathSciNet review: 2159760