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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Real $k$-flats tangent to quadrics in $\mathbb{R} ^n$


Authors: Frank Sottile and Thorsten Theobald
Journal: Proc. Amer. Math. Soc. 133 (2005), 2835-2844
MSC (2000): Primary 14N10, 51M30, 14P99, 52C45, 05A19
Published electronically: April 8, 2005
MathSciNet review: 2159760
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Abstract: Let $d_{k,n}$ and $\char93 _{k,n}$ denote the dimension and the degree of the Grassmannian $\mathbb{G} _{k,n}$, respectively. For each $1 \le k \le n-2$ there are $2^{d_{k,n}} \cdot \char93 _{k,n}$ (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.


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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: http://dx.doi.org/10.1090/S0002-9939-05-07880-9
PII: S 0002-9939(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.