Real -flats tangent to quadrics in

Authors:
Frank Sottile and Thorsten Theobald

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

Published electronically:
April 8, 2005

MathSciNet review:
2159760

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let and denote the dimension and the degree of the Grassmannian , respectively. For each there are (a priori complex) -planes in tangent to general quadratic hypersurfaces in . We show that this class of enumerative problems is fully real, i.e., for there exists a configuration of real quadrics in (affine) real space so that all the mutually tangent -flats are real.

**1.**P. Aluffi and W. Fulton.

Lines tangent to four surfaces containing a curve,

in preparation.**2.**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**, https://doi.org/10.2307/3062151**3.**W. Fulton.*Introduction to Intersection Theory in Algebraic Geometry*.

CBMS Regional Conference Series in Mathematics, vol. 54, AMS, Providence (RI), 3rd printing, 1996. MR**0735435 (85j:14008)****4.**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**

W. V. D. Hodge and D. Pedoe,*Methods of algebraic geometry. Vol. II. Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties*, Cambridge, at the University Press, 1952. MR**0048065****5.**S. L. Kleiman and Dan Laksov,*Schubert calculus*, Amer. Math. Monthly**79**(1972), 1061–1082. MR**0323796**, https://doi.org/10.2307/2317421**6.**I. G. Macdonald, J. Pach, and T. Theobald,*Common tangents to four unit balls in ℝ³*, Discrete Comput. Geom.**26**(2001), no. 1, 1–17. MR**1832726**, https://doi.org/10.1007/s004540010090**7.**H. Schubert. Anzahlbestimmungen für lineare Räume beliebiger Dimension.*Acta Math.*8:97-118, 1886.**8.**V. Sedykh and B. Shapiro. Two conjectures on convex curves. Preprint, 2002.`math.AG/``0208218`.**9.**Frank Sottile,*Enumerative geometry for the real Grassmannian of lines in projective space*, Duke Math. J.**87**(1997), no. 1, 59–85. MR**1440063**, https://doi.org/10.1215/S0012-7094-97-08703-2**10.**Frank Sottile,*The special Schubert calculus is real*, Electron. Res. Announc. Amer. Math. Soc.**5**(1999), 35–39. MR**1679451**, https://doi.org/10.1090/S1079-6762-99-00058-X**11.**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**, https://doi.org/10.1007/978-3-662-04851-1_6**12.**Frank Sottile and Thorsten Theobald,*Lines tangent to 2𝑛-2 spheres in ℝⁿ*, Trans. Amer. Math. Soc.**354**(2002), no. 12, 4815–4829. MR**1926838**, https://doi.org/10.1090/S0002-9947-02-03014-3**13.**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****14.**R. Vakil.

Schubert Induction. Preprint, 2003.`math.AG/0302296`.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
14N10,
51M30,
14P99,
52C45,
05A19

Retrieve articles in all journals with MSC (2000): 14N10, 51M30, 14P99, 52C45, 05A19

Additional Information

**Frank Sottile**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843

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

Email:
theobald@math.tu-berlin.de

DOI:
https://doi.org/10.1090/S0002-9939-05-07880-9

Keywords:
Tangents,
transversals,
quadrics,
enumerative geometry,
real solutions,
Grassmannian

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

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.