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Real -flats tangent to quadrics in
Authors:Frank Sottile   Thorsten Theobald
Affiliation:Department of Mathematics, Texas A&M University, College Station, Texas 77843 ; Institut für Mathematik, MA 6-2, Technische Universität Berlin, Strasse des 17. Juni 1936, D-10623 Berlin, Germany
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.

Keywords:Tangents   transversals   quadrics   enumerative geometry   real solutions   Grassmannian
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