\documentclass[12pt]{amsart} \headheight=8pt \topmargin=0pt \textheight=624pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \newif\ifpictures \picturestrue % picturestrue: incl. pictures % picturesfalse: without pictures, faster complilation \usepackage{amsmath} \usepackage{amssymb} \usepackage{latexsym} \usepackage[dvips]{graphicx} \ifpictures \input{epsf} \fi \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} %\newtheorem{definition}[theorem]{Definition} \newtheorem{fact}[theorem]{Fact} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{conjecture}[theorem]{Conjecture} %\newenvironment{proof}{ \noindent {\bf{Proof.}} }{ \hfill $\Box$\medskip } \newenvironment{definition}{ \noindent {\bf{Definition.}} }{} \newenvironment{claim}{ \noindent {\bf{Claim.}} }{} \newenvironment{remark}{ \noindent {\bf{Remark.}} }{ \medskip } \def\problem#1#2#3{ \medskip \noindent {\bf Problem {#1}:} \medskip \noindent \begin{tabular}{@{}lp{12cm}} {\bf Instance:} & {#2}. \\ {\bf Question:} & {#3}. \\ \end{tabular} \bigskip } \newcommand{\BB}{\mathbb{B}} %\newcommand{\RR}{\mathbb{R}} \newcommand{\R}{\mathbb{R}} \newcommand{\CC}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} %\newcommand{\PP}{\mathbb{P}} \renewcommand{\P}{\mathbb{P}} \newcommand{\NP}{{\mathbb N \mathbb P}} \newcommand{\BTWO}{\mathcal{B}_2} \newcommand{\PV}{\mathcal{P}_{\mathcal{V}}} \newcommand{\PH}{\mathcal{P}_{\mathcal{H}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\def\publname{{\today} \def\publname{{\quad} %Appeared in \emph{Really Good Journal} (2003)} \def\currentvolume{} \def\currentissue{} % for some reason it can't get the date info from here %\issueinfo{X}{3}{January}{2003} \pagespan{1}{1} \PII{}} \copyrightinfo{}{} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sloppy \begin{document} % ----------------------------------------------------------------- % Begin text % ----------------------------------------------------------------- \title{Real $k$-flats tangent to quadrics in $\R^n$} \author{Frank Sottile} \address{Mathematical Sciences Research Institute\\ 17 Gauss Way\\ Berkeley, CA 94720\\ USA} \email{sottile@msri.org} \urladdr{http://www.math.umass.edu/\~{}sottile} \author{Thorsten Theobald} \address{Zentrum Mathematik\\ Technische Universit\"at M\"unchen\\ D--85747 Garching bei M\"unchen\\ Germany, and Mathematical Sciences Research Institute\\ 17 Gauss Way\\ Berkeley, CA 94720\\ USA} \email{theobald@ma.tum.de} \urladdr{http://www-m9.ma.tum.de/\~{}theobald/} %\date{June 26, 2002} \thanks{Research of Sottile supported by NSF grant DMS-0070494 and the Clay Mathematical Institute} \keywords{ tangents, transversals, quadrics, enumerative geometry, real solutions, \\ \indent \hspace{113pt}Grassmannian.\\ \indent \emph{2000 Mathematics Subject Classification} 14N10, 51M30, 14P99, 52C45, 05A19. } % in 2000 Math. Subject Classification: %14N10: Enumerative problems (combinatorial problems) %51M30 line geometries and their generalizations %14P99: Real algebraic and real analytic geometry; none of the above, but in % this section %52C45: Combinatorial complexity of geometric structures %05A19: enumerative combinatorics; combinatorial identities %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{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 $\P^n$ tangent to $d_{k,n}$ general quadratic hypersurfaces in $\P^n$. We show that this class of enumerative problem 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 $\R^n$ so that all the mutually tangent $k$-flats are real. \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Introduction} Understanding the real solutions of a system of polynomial equations is a fundamental problem in mathematics (see, e.g., \cite{sturmfels-cbms} for some recent lines of research and applications). However, as pointed out in \cite[p.~55]{fulton-b96}, even for problem classes with a finite number of complex solutions (enumerative problems), the question of how many solutions can be real is still widely open. %(see also the recent survey \cite{sottile-erag}). A class of enumerative problems is called \emph{fully real} if there are general real instances for which all the (a priori complex) solutions are real. One of us (Sottile) began a systematic study of this question in the \emph{special Schubert calculus}~\cite{So97a,So99a}, a class of enumerative problems from classical algebraic geometry. This special Schubert calculus asks for linear subspaces of a fixed dimension meeting some given (general) linear subspaces (whose dimensions and number ensure a finite number of solutions) in $n$-dimensional complex projective space $\mathbb{P}^n$. For any given dimensions of the subspaces, this problem is fully real, i.e., there exist \emph{real} linear subspaces for which each of the a priori complex solutions are \emph{real}. In particular, for $1 \le k \le n-2$ there are $d_{k,n}:=(k+1)(n-k)$ real $(n{-}k{-}1)$-planes $U_{1}, \ldots, U_{d_{k,n}}$ in $\mathbb{P}^n$ with \[ \#_{k,n} := \frac{1!2!\cdots k!((k+1)(n-k))!}{(n-k)!(n-k+1)! \cdots n!} \] real $k$-planes meeting $U_1, \ldots, U_{d_{k,n}}$. Here, $d_{k,n}$ and $\#_{k,n}$ are the dimension and the degree of the Grassmannian $\mathbb{G}_{k,n}$, respectively (see~\cite{kleiman-laksov-72,schubert-1886}). These were the first results showing that a large class of non-trivial enumerative problems is fully real. We continue this line of research by considering $k$-flats tangent to quadratic hypersurfaces (hereafter \emph{quadrics}). This is also motivated by recent investigations in computational geometry (see~\cite{MPT01,M2,STh02}). It was shown in~\cite{STh02} that $2n{-}2$ general \emph{spheres} in affine real space $\mathbb{R}^n$ have at most $3\cdot 2^{n-1}$ common tangent lines in $\mathbb{C}^n$, and that there exist spheres for which all the a priori complex tangent lines are \emph{real}. The present paper addresses the following question: What is the maximum number of real $k$-flats simultaneously tangent to $d_{k,n}$ general quadrics in $\mathbb{R}^n$ (respectively in $\mathbb{P}_{\mathbb{R}}^n$)? As this problem may be formulated as the complete intersection of $d_{k,n}$ quadratic equations on the Grassmannian of $k$-planes in $\P^n$, the expected number of complex solutions is the product of the degrees of the equations with the degree of the Grassmannian, i.e., $ 2^{d_{k,n}}\cdot \#_{k,n}$. We show that the problem is fully real: \begin{theorem}\label{T:allreal} Let $1 \le k \le n-2$. Given $d_{k,n}$ general quadrics in $\P^n$ there are $2^{d_{k,n}} \cdot \#_{k,n}$ complex $k$-planes that are simultaneously tangent to all $d_{k,n}$ quadrics. Furthermore, there is a choice of quadrics in $\R^n$ for which all the $k$-flats are real, distinct, and lie in affine space $\R^n$. \end{theorem} When $k=1$, we have $d_{1,n} = 2(n-1)$ and $\#_{1,n}$ is the Catalan number $\#_{1,n} = \frac{1}{n} \binom{2n-2}{n-1}$. The following table exhibits the large discrepancy between the number of lines tangent to spheres and the number of lines tangent to general quadrics. When $n=3$ this discrepancy was accounted for by Aluffi and Fulton~\cite{AF}. \[ \begin{tabular}{|c||c|c|c|c|c|c|c|}\hline $n$& 3 & 4 & 5 & 6 & 7 & 8 & 9\\\hline\hline \rule{0pt}{13pt} $3\cdot 2^{n-1}$&12&24&48&96&192&384&768\\\hline\rule{0pt}{12pt} $2^{d_{1,n}} \cdot \#_{1,n}$&32&320&3584&43008&540672&7028736&93716480\\\hline \end{tabular} \] In Section~\ref{S:plucker}, we review some facts on Pl\"ucker coordinates of $k$-planes in projective space. In Section~\ref{S:proof}, we combine recent results in the real Schubert calculus with classical perturbation arguments adapted to the real numbers to prove Theorem~\ref{T:allreal}. Since the proof for general $(k,n)$ is non-constructive, we give a symbolic, constructive proof for the case $(k,n) = (1,3)$ in Section~\ref{S:symbolic}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Preliminaries}\label{S:plucker} We review the well-known \emph{Pl\"ucker coordinates} of $k$-dimensional linear subspaces (hereafter $k$-planes) in complex projective space $\P^n$ (see, e.g., \cite{hodge-pedoe}). Let $U$ be a $k$-plane in $\P^n$ which is spanned by the columns of an $(n+1) \times (k+1)$-matrix $L$. For every subset $I \subset \{0, \ldots, n\}$ of size $k+1$ let $p_{I}$ be the $(k+1) \times (k+1)$-subdeterminant of $L$ given by the rows in $I$ and let $N:=\binom{n+1}{k+1}-1$. Then $p := (p_I)_{I \subset \{0, \ldots, n\}, |I| = k+1} \in \P^N$ is the \emph{Pl\"ucker coordinate} of $U$. The set of all $k$-planes in $\P^n$ is called the \emph{Grassmannian of $k$-planes in} $\P^n$ and is denoted by $\mathbb{G}_{k,n}$. If the indices are written as ordered tuples then the Pl\"ucker coordinates are skew-symmetric in the indices. $\mathbb{G}_{k,n}$ is in 1-1-correspondence with the set of vectors in $\P^N$ satisfying the \emph{Pl\"ucker relations}, i.e., \begin{equation} \label{eq:plueckerrelation} \sum_{l=1}^{k+1} (-1)^l p_{i_1 \ldots \hat{i_l} \ldots i_{k+1}} \, p_{j_1 \ldots j_{k-1} i_l} \ =\ 0 \end{equation} for every $I=\{i_1, \ldots, i_{k+1}\}$, $J = \{j_1 , \ldots, j_{k-1}\} \subset \{0, \ldots, n\}$ of strictly ordered index sets (where $\hat{\hspace*{1mm}}$ over an index means that it is omitted). See, e.g., \cite[\S VII.6]{hodge-pedoe}. By Schubert's results \cite{schubert-1886}, the dimension of $\mathbb{G}_{k,n}$ is $d_{k,n} = (k+1)(n-k)$ and its degree is $\#_{k,n}$. If an $(n{-}k{-}1)$-plane $V$ is given as the intersection of the $k+1$ hyperplanes $\sum_{i=0}^n v_i^{(0)} x_i {~=~} 0, \ldots , \sum_{i=0}^n v_i^{(k)} x_i {~=~} 0$, then the \emph{dual Pl\"ucker coordinate} $q = (q_I)_{I \subset \{0, \ldots, n\}, |I| = k+1} \in \P^N$ of $V$ is defined by the $(k+1) \times (k+1)$-subdeterminants of the matrix with columns $v^{(0)}, \ldots, v^{(k)}$. A $k$-plane $U$ intersects an $(n{-}k{-}1)$-plane $V$ in $\P^n$ if and only if the dot product of the Pl\"ucker coordinate $p$ of $U$ and the dual Pl\"ucker coordinate $q$ of $V$ vanishes, i.e., if and only if \begin{equation} \label{eq:transversalcond} \sum_{I \subset \{0, \ldots, n\}, |I| = k+1} p_{I} q_{I} \ = \ 0 \end{equation} (see, e.g., \cite[Theorem~VII.5.I]{hodge-pedoe}). %Since this is a linear equation in the Pl\"ucker coordinates of the %subspace $U$, the set of $k$-planes intersecting a given %$(n{-}k{-}1)$-plane is described by a hyperplane section of the %Grassmannian $\mathbb{G}_{k,n}$. We use Pl\"ucker coordinates to characterize the $k$-planes tangent to a given quadric in $\mathbb{P}^n$ (see \cite{M2}). We identify a quadric $x^{\mathrm{T}} Q x = 0$ in $\P^n$ with its symmetric $(n{+}1)\times(n{+}1)$-representation matrix $Q$. Further, for $r \in \N$ let $\wedge^r$ denote the $r$-th exterior power of matrices % \begin{eqnarray*} \wedge^r \ \colon\ \mathbb{C}^{m\times n} & \to & \mathbb{C}^{\binom{m}{2}\times\binom{n}{2}} \end{eqnarray*} % (see~\cite{M2}). The row and column indices of the resulting matrix are subsets of cardinality~$r$ of $\{1, \ldots, m\}$ and $\{1, \ldots, n\}$, respectively. For $I \subset \{1, \ldots, m\}$ with $|I| = r$ and $J \subset \{1, \ldots, n\}$ with $|J| = r$, $\bigl(\wedge^r A\bigr)_{I,J}$ is the subdeterminant of $A$ whose rows are indexed by $I$ and whose columns are indexed by $J$. If a $k$-plane $U \subset \P^n$ is spanned by the columns of an $(n{+}1)\times (k+1)$-matrix $L$, then the $\binom{n+1}{k+1} \times 1$-matrix $\wedge^{k+1} L$, considered as a vector in $\P^N$, is the Pl\"ucker coordinate of $U$. Recall the following algebraic characterization of tangency: A $k$-plane $U$ is tangent to a quadric $Q$ if the restriction of the quadratic form to $U$ is singular (which includes the case $U \subset Q$). When the quadric is smooth, this implies that $U$ is tangent to the quadric in the usual geometric sense. \begin{proposition} [Proposition 5.5.3 of~\cite{M2}] \label{le:quadrictangentcond} A $k$-plane $U \subset \P^n$ is tangent to a quadric $Q$ if and only if the Pl\"ucker coordinate $p_U$ of $U$ satisfies \begin{equation} \label{eq:quadrictangentcond} p_U^{\mathrm{T}} \, \bigl(\wedge^{k+1} Q\bigr) \, p_U = 0 \, . \end{equation} \end{proposition} A $k$-\emph{flat} in affine real space $\R^n$ is a $k$-dimensional affine subspace in $\R^n$. Throughout the paper we assume that $\R^n$ is naturally embedded in the real projective space $\P_{\mathbb{R}}^n$ via $(x_1, \ldots, x_n) \mapsto (1,x_1, \ldots, x_n) \in \P_{\mathbb{R}}^n$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Proof of the main theorem}\label{S:proof} We first illustrate the essential geometric idea underlying our constructions for $(k,n) = (1,3)$, which is the first nontrivial case. Here, Theorem~\ref{T:allreal} states that there exists a configuration of four quadrics in $\R^3$ with $32$ distinct real common tangent lines. By~\eqref{eq:transversalcond}, the set of lines meeting four given lines in $\P^3$ is the intersection of four hyperplanes on the Grassmannian $\mathbb{G}_{1,3}$, and hence there are at most two or infinitely many common lines meeting $\ell_1, \ldots, \ell_4$. If $e_1$ and $e_2$ are opposite edges in a tetrahedron $\Delta \subset \R^3$, then the lines underlying $e_1$ and $e_2$ are the two common transversals of the four lines underlying the other four edges (see~Figure~\ref{fi:4lines2transversals}). \ifpictures \begin{figure}[htb] \[ \unitlength1.2pt \begin{array}{c@{\hspace*{1.5cm}}c} \begin{picture}(150,130)(0,12) \put(3,3){\epsfxsize=168pt\epsfbox{figures/4lines2transversals.eps}} \put( 41, 7){$\ell_1$} \put( 85, 40){$\ell_2$} \put( 8, 96){$\ell_3$} \put( 50,105){$\ell_4$} \put( 35, 55){$e_1$} \put( 98, 93){$e_2$} \end{picture} & \\ [-5.8cm] & \epsfxsize=2.4in \epsfbox{figures/realquad32.eps} \\ \end{array} \] \caption{A tetrahedron configuration of four lines in $\R^3$ with two real transversals and a configuration of four quadrics with 32 real tangents.} \label{fi:4lines2transversals} \end{figure} \fi Consider the lines $\ell_1, \ldots, \ell_4$ as (degenerate) infinite circular cylinders with radius $r=0$. When the radius is slightly increased, then the cylinders intersect pairwise in the regions (combinatorially) given by the four vertices of $\Delta$, and the common tangents roughly have the direction of $e_1$ or $e_2$. Since the neighborhood of a vertex is divided into four regions by the two cylinders, and since each region contains common tangents, this gives $4 \cdot 4$ tangents close to the direction of $e_1$ and $4 \cdot 4$ tangents close to the direction of $e_2$ (see Figure~\ref{fi:4lines2transversals}.) For the general case, let $1 \le k \le n-2$. By Section~\ref{S:plucker}, the number of $k$-planes in $\P^n$ simultaneously meeting $d_{k,n}$ general $(n{-}k{-}1)$-planes is $\#_{k,n}$. We begin with a configuration of $d_{k,n}$ real $(n{-}k{-}1)$-flats $U_1, \ldots, U_{d_{k,n}}$ in $\R^n$ having $\#_{k,n}$ real $(n{-}k{-}1)$-flats simultaneously meeting $U_1, \ldots, U_{d_{k,n}}$. We then argue that we can replace each of these $(n{-}k{-}1)$-flats by a real quadric such that for each of the $k$-flats, there are $2^{d_{k,n}}$ nearby real $k$-flats tangent to each quadric. \begin{proposition} \label{pr:realflats} For $1 \le k \le n-2$, there exists a configuration of $d_{k,n}$ real $(n{-}k{-}1)$-flats $U_1, \ldots, U_{d_{k,n}}$ in $\R^n$ such that there exist exactly $\#_{k,n}$ real $k$-flats simultaneously meeting $U_1, \ldots, U_{d_{k,n}}$. \end{proposition} \begin{proof} The corresponding statement for real projective space $\P_{\R}^n$ was proven for $k=1$ in~\cite[Theorem~C]{So97a} and for $k\ge 2$ in~\cite{So99a}. We deduce the affine counterpart above simply by removing a real hyperplane that contains none of the $(n{-}k{-}1)$-flats or any of the transversal $k$-flats. \end{proof} For $k=1$, the purely existential statement in~\cite{So97a} and Proposition~\ref{pr:realflats} was improved by Eremenko and Gabrielov~\cite{EG02} who gave the following explicit construction of such a collection of $(n{-}2)$-flats. Let $\gamma:\R \to \R^n$, $\gamma(s)\ :=\ (1,s,s^2,\dotsc,s^{n-1})^{\mathrm{T}}$ be the moment curve in $\R^n$. For each $s\in\R$, set $U(s)$ to be \[ U(s)\ :=\ \textrm{linear span}\bigl(\gamma(s),\gamma'(s),\dotsc, \gamma^{(n{-}3)}(s)\bigr)\,. \] Geometrically, $U(s)$ is the $(n{-}2)$-flat osculating the moment curve at the point $\gamma(s)$. By \cite{EG02}, for any distinct $s_1,\ldots,s_{2n-2}\in\R$, the $(n{-}2)$-flats $U(s_1), U(s_2),\dotsc, U(s_{2n-2})$ have exactly $\#_{1,n} = C_{n-1}$ common real transversals, where $C_{n} = \frac{1}{n+1} \binom{2n}{n}$ is the $n$-th Catalan number. For general $k$, it is only known that there \emph{exist} distinct $s_1, \ldots, s_{d_{k,n}} \in \R$ such that there are $\#_{k,n}$ distinct real $k$-flats meeting the osculating $(n{-}k{-}1)$-flats to the moment curve at $s_1, \ldots, s_{d_{k,n}}$ \cite{So99a}. The \emph{conjecture on total reality} in \cite[\S 1 and \S 4]{sedykh-shapiro-2002} conjectures that any choice of distinct $s_1, \ldots, s_{d_{k,n}} \in \R$ implies reality of all transversal subspaces. \medskip \begin{definition} {\rm Suppose that $1 \le k\le n-2$, and let $U \subset \R^n$ be a $k$-flat and $r > 0$. The {\it $k$-cylinder} $\textrm{Cy}(U,r)$ is the set of points having Euclidean distance $r$ from $U$. } \end{definition} \medskip This quadratic hypersurface is smooth in $\R^n$ but its extension to $\P^n$ is singular. A $k'$-flat $V \subset \R^n$ is tangent to $\textrm{Cy}(U,r)$ if and only if its Euclidean distance to $U$ is $r$. We will use the following basic property of intersection multiplicities~\cite[p.~1]{fulton-b96}. \begin{proposition}\label{pr:multiplicity} Let $\mathcal{A}$ be an algebraic curve in $\P^n$, and let $x$ be a singular point on $\mathcal{A}$. For any hyperplane $H \subset \P^n$ such that $x$ is an isolated point in $\mathcal{A} \cap H$, the intersection multiplicity of $\mathcal{A}$ and $H$ in $x$ is greater than $1$. \end{proposition} \begin{theorem} \label{th:inductive} Let $1 \le k \le n-2$, and let $U_1,U_2,\ldots,U_{d_{k,n}}$ be $(n{-}k{-1})$-flats in $\R^n$ having exactly $\#_{k,n}$ common transversal $k$-flats, all real. For each $i=0,1,\dotsc, d_{k,n}$, there exist $r_1,\dotsc,r_{i} > 0$ such that there are exactly $2^i \cdot \#_{k,n}\,$ distinct $k$-flats, each of them real, that are simultaneously tangent to each of the $(n{-}k{-}1)$-cylinders ${\rm{Cy}}(U_j,r_j)$, $j=1,\dotsc,i$, and also meet each of the $(n{-}k{-}1)$-flats $U_{i+1},\dotsc,U_{d_{k,n}}$. \end{theorem} The case of $i=d_{k,n}$ implies Theorem~\ref{T:allreal}. \begin{proof} We induct on $i$, with the case of $i=0$ being the hypothesis of the theorem. Suppose that $i\leq d_{k,n}$ and that there exist $r_1,\dotsc,r_{i-1} > 0$ such that there are exactly $2^{i-1} \cdot \#_{k,n} \,$ distinct $k$-flats $V_1, \ldots, V_{2^{i-1} \#_{k,n}}$ which are simultaneously tangent to $\mathrm{Cy}(U_j, r_j)$ for each $j=1, \ldots, i-1$, and meet each of $U_{i}, \ldots, U_{d_{k,n}}$, and each of these $k$-flats is real. Now we drop the condition that the $k$-flats meet $U_i$. Let $\mathcal{A} \subset \mathbb{G}_{k,n}$ be the curve of $k$-flats that are tangent to the cylinders $\textrm{Cy}(U_j,r_j)$ for $j=1,\dotsc,i{-}1$ and that also meet each of the $(n{-}k{-}1)$-flats $U_{i+1},\dotsc,U_{d_{k,n}}$. Since $\mathcal{A}$ is the intersection of $i{-}1$ quadrics (the tangency conditions) with $d_{k,n}-i$ hyperplanes (conditions to meet the remaining $U_j$) on the Grassmannian, it has degree at most $2^{i-1}d_{k,n}$. Since its intersection with the hyperplane defined by $U_i$ consists of $2^{i-1} d_{k,n}$ points, we conclude that the degree of $\mathcal{A}$ is $2^{i-1} d_{k,n}$ and (by Proposition~\ref{pr:multiplicity}) that each of these points is a smooth point of $\mathcal{A}$. Let $V \in \{V_1, \ldots, V_{2^{i-1} \#_{k,n} }\}$. Since $V$ is a smooth real point of the real curve $\mathcal{A} \subset \mathbb{G}_{k,n}$ (i.e., $V \subset \P_{\R}^n$), the real points of $\mathcal{A}$ contain a smooth arc $\alpha$ containing $V$ with $\alpha \cap (\{V_1,\dotsc,V_{2^{i-1} \#_{k,n}}\} \setminus V) = \emptyset$. Let $\varphi\colon(-\delta,\delta)\to\alpha$ be a smooth parametrization of the arc $\alpha$ with $\varphi(0)=V$. Such a parametrization exists, for example, by the Implicit Function Theorem. Thus, for $t\in(-\delta,\delta)\setminus \{0\}$, the real $k$-flat $\varphi(t)$ does not meet $U_i$ and so it has a positive Euclidean distance $d(t)$ from $U_i$. Since $d(t)$ is a continuous function of $t$, for $\rho \in \R$ with $0 < \rho < \min\{d(-\delta/2),d(\delta/2)\}$ there are at least two distinct real $k$-flats in $\alpha$ whose Euclidean distance to $U_i$ is $\rho$. In this way, we obtain $2^{i-1} \cdot \#_{k,n}$ such arcs, each containing one of $V_1,\dotsc,V_{2^{i-1} \#_{k,n}}$. We may assume that these arcs are pairwise disjoint. Let $00$, these quadrics are deformations of the lines with rank~4 and signature~0---smooth ruled surfaces. \begin{theorem} \label{th:32construction} Let $(\alpha,\beta) \in \R^2$ satisfy \[ \alpha\beta(1-\alpha\beta)(1-\beta^2)(1-\alpha^2) \bigl((1-\alpha)^2(1-\beta)^2-16\alpha\beta\bigr) \ \neq \ 0\, . \] Then there are $32$ distinct (possibly complex) common tangent lines to $Q_1, \ldots, Q_4$. If\/ $0<\alpha,\beta<3 - 2 \sqrt{2}$, then each of these $32$ tangent lines are real. \end{theorem} \begin{proof} Since the quadrics only contain monomials of the form $x_i^2$, the tangent equations~\eqref{eq:quadrictangentcond} of $Q_1, \ldots, Q_4$ only contain monomials of the form $p_{ij}^2$. Hence, the four tangent equations give the following system of linear equations in $p_{01}^2, \ldots, p_{23}^2$: \[ \left( \begin{matrix} -\beta &-\beta & 1 & \beta^2 & - \beta & - \beta \\ 1 &-\beta & -\beta & -\beta & - \beta & \beta^2 \\ -\alpha &-\alpha & \alpha^2 & 1 & - \alpha & - \alpha \\ \alpha^2 &-\alpha & -\alpha & -\alpha & - \alpha & 1 \end{matrix} \right) \left( \begin{matrix} p_{01}^2 \\ p_{02}^2 \\ p_{03}^2 \\ p_{12}^2 \\ p_{13}^2 \\ p_{23}^2 \end{matrix} \right) \ =\ 0 \, . \] Permute the variables into the order $(p_{02},p_{13},p_{03},p_{12},p_{01},p_{23})$. For $\alpha,\beta$ satisfying % \begin{equation}\label{E:abnez} \alpha\beta(1-\alpha\beta)(1+\beta)(1+\alpha)\ \neq \ 0\,, \end{equation} % Gaussian elimination yields the following system: \[ \left( \begin{matrix} -\beta & -\beta & (1-\alpha)(1-\beta) & 0 & 0 & 0 \\ 0 & 0 & \alpha & -\beta & 0 & 0 \\ 0 & 0 & 0 & -\beta & \alpha & 0 \\ 0 & 0 & 0 & 0 & \alpha & -\beta \end{matrix} \right) \left( \begin{matrix} p_{02}^2 \\ p_{13}^2 \\ p_{03}^2 \\ p_{12}^2 \\ p_{01}^2 \\ p_{23}^2 \end{matrix} \right) \ =\ 0 \, . \] Together with the Pl\"ucker equation~\eqref{eq:plueckerrelation}, this gives the following system of equations: % \begin{eqnarray} - \beta p_{02}^2 - \beta p_{13}^2 + (1-\alpha)(1-\beta) p_{03}^2 & = & 0 \, , \label{eq:eq1} \\ p_{01} p_{23} - p_{02} p_{13} + p_{03} p_{12} & = & 0 \, , \label{eq:eq2} \\ \alpha p_{01}^2 \ =\ \alpha p_{03}^2 \ =\ \beta p_{12}^2& = & \beta p_{23}^2 \, . \label{eq:eq3} \end{eqnarray} % For $\alpha,\beta$ satisfying~\eqref{E:abnez} as well as $(1-\alpha)(1-\beta)\neq 0$, we distinguish the following three disjoint cases. \medskip \noindent \emph{Case 1:} $p_{02} = 0$. \noindent Since $p_{13} = 0$ would imply that all components are zero and hence contradict $(p_{01}, \ldots, p_{23})^{\mathrm{T}} \in \P^5$, we can assume $p_{13} = 1$. Then~\eqref{eq:eq1} and (\ref{eq:eq3}) imply \[ \alpha p_{01}^2\ =\ \alpha p_{03}^2\ =\ \beta p_{12}^2\ =\ \beta p_{23}^2\ =\ \frac{\alpha\beta}{(1-\alpha)(1-\beta)} \neq 0 \, . \] Since (\ref{eq:eq2}) implies $\mathrm{sgn}(p_{01} p_{23}) = -\mathrm{sgn}(p_{03} p_{12})$, only 8 of the $2^4=16$ sign combinations for $p_{01}, p_{03}, p_{12}, p_{23}$ are possible. Namely, the 8 (complex) solutions for $p_{01}, p_{03},p_{12}, p_{23}$ are \begin{equation} \label{eq:solutions4tuple} (p_{01},p_{03},p_{12},p_{23})^{\mathrm{T}} = \frac{1}{\sqrt{(1-\alpha)(1-\beta)}} \, (\gamma_{01}\beta, \gamma_{03}\beta, \gamma_{12}\alpha, - \mathrm{sgn}(\gamma_{01} \gamma_{03} \gamma_{12})\alpha)^{\mathrm{T}} \end{equation} with $\gamma_{01}, \gamma_{03}, \gamma_{12} \in \{-1,1\}$. Hence, for $\alpha,\beta \in \R^2$ satisfying~\eqref{E:abnez}, this case gives 8 distinct common tangents. \medskip \noindent \emph{Case 2:} $p_{13} = 0$. \\ This case is symmetric to case 1. Setting $p_{13} = 1$, the resulting 8 solutions for the variables $p_{01}, p_{03}, p_{12}, p_{23}$ are the same ones as in~(\ref{eq:solutions4tuple}). \medskip \noindent \emph{Case 3:} $p_{02} p_{13} \neq 0$. \\ Without loss of generality, we can assume $p_{02} = 1$. Solving (\ref{eq:eq2}) for $p_{13}$ and substituting this expression into $(\ref{eq:eq1})$ yields \[ - \beta - \beta p_{01}^2 p_{23}^2 - \beta p_{03}^2 p _{12}^2 - 2 \beta p_{01} p_{03} p_{12} p_{23} + (1-\alpha)(1-\beta) p_{03}^2\ =\ 0 \, . \] We use~\eqref{eq:eq3} to write this in terms of $p_{01}$. This is straightforward for the squared terms, but for the other terms, we observe that, by~\eqref{eq:eq3}, $p_{01}p_{23}=\pm p_{03}p_{12}$ and since $p_{02} p_{13} \neq 0$, the Pl\"ucker equation~\eqref{eq:eq2} implies these have the same sign. This gives the quartic equation in $p_{01}$ \[ - \beta + (1-\alpha)(1-\beta) p_{01}^2 - 4 \alpha p_{01}^4\ =\ 0 \, , \] whose discriminant is % \begin{equation}\label{E:disc} (1-\alpha)^2(1-\beta)^2-16\alpha\beta \, . \end{equation} % Hence, for $\alpha,\beta \in \R^2$ satisfying~\eqref{E:abnez}, and for which this discriminant does not vanish, there are two different solutions for $p_{01}^2$. For each of these two solutions for $p_{01}^2$, there are 8 distinct solutions for $p_{01}, p_{03}, p_{12}, p_{23}$, namely \begin{equation} \label{eq:solutions4tuplecase3} (p_{01},p_{03},p_{12},p_{23})^{\mathrm{T}} \ =\ \sqrt{ p_{01}^2} \, ( \gamma_{01}, \gamma_{03}, \gamma_{12}, \mathrm{sgn}(\gamma_{01} \gamma_{03} \gamma_{12}))^{\mathrm{T}} \end{equation} with $\gamma_{01}, \gamma_{03}, \gamma_{12} \in \{-1,1\}$. Since $p_{13}$ is uniquely determined by $p_{01}$, $p_{02}$, $p_{03}$, $p_{12}$, case~3 gives 16 distinct common tangents.\medskip In order to determine when all solutions are real, suppose first that $\alpha=\beta$. Then the discriminant~\eqref{E:disc} becomes $(\alpha^2-6\alpha+1)(\alpha+1)^2$, and its smallest positive root is $\alpha_0 := 3 - 2 \sqrt{2} \approx 0.17157$. In particular, for $0 < \alpha < \alpha_0$, the discriminant in case~3 is positive and both solutions for $p_{01}^2$ are positive. Thus, for $0 < \beta=\alpha < \alpha_0$, the solutions of all three cases are distinct and real. Next, fix $0 < \alpha < \alpha_0$ and suppose that $0 < \beta<\alpha$. Then the discriminant~\eqref{E:disc} is positive: for fixed $0 < \alpha < \alpha_0$, the discriminant~\eqref{E:disc} is decreasing in $\beta$ for $0 < \beta < \alpha$ and positive when $\beta = \alpha$. This concludes the proof of Theorem~\ref{th:32construction}. \end{proof} Figure~\ref{fi:explproj} illustrates the construction and the 32 tangents for $\alpha=1/10$ and $\beta=1/20$. \ifpictures \begin{figure}[htb] \[ \epsfxsize=3.3in \epsfbox{figures/fig3.eps} \] \caption{The configuration of quadrics from Theorem~\ref{th:32construction}.} \label{fi:explproj} \end{figure} \fi \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } % \MRhref is called by the amsart/book/proc definition of \MR. \providecommand{\MRhref}[2]{% \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \begin{thebibliography}{10} \bibitem{AF} P.~Aluffi and W.~Fulton. \newblock Lines tangent to four surfaces containing a curve, %2001, \newblock in preparation. %\bibitem{bowen-wang-b76} %R.M.~Bowen and C.-C.~Wang. \emph{Introduction to vectors and tensors}, vol.~1, % Plenum Press, New York, 1976. %\bibitem{clo-b96} %D.~Cox, J.~Little, and D.~O'Shea. %\newblock {\em Ideals, Varieties, and Algorithms}, 2nd edition. %\newblock UTM. Springer-Verlag, New York, 1996. \bibitem{EG02} A.~Eremenko and A.~Gabrielov. \newblock Rational functions with real critical points and the {B}.~and {M}.~{S}hapiro conjecture in real enumerative geometry. \newblock {\em Ann.\ Math.} 155:131--156, 2002. %\bibitem{fischer-piontkowski-b2001} %G.~Fischer and J.~Piontkowski. %\newblock {\em Ruled Varieties.} %Vieweg, Braunschweig, 2001. \bibitem{fulton-b96} W.~Fulton. \newblock {\em Introduction to Intersection Theory in Algebraic Geometry}. \newblock CBMS Regional Conference Series in Mathematics, vol.~54, AMS, Providence (RI), 3rd printing, 1996. %\bibitem{golab-b74} %S.~Golab. \emph{Tensor calculus}, Elsevier, Amsterdam, 1974. \bibitem{hodge-pedoe} W.V.D. Hodge and D.~Pedoe. \newblock {\em Methods of Algebraic Geometry}, vols.~I and~II. \newblock Cambridge Univ.~Press, 1947 and 1952. %\bibitem{kendig-b77} %K.~Kendig. %\newblock {\em Elementary Algebraic Geometry}, %\newblock Graduate Texts in Mathematics, vol.~44, % Springer-Verlag, New York, 1977. \bibitem{kleiman-laksov-72} S.~Kleiman and D.~Laksov. \newblock Schubert calculus, \newblock {\em Amer.\ Math.\ Monthly} 79:1061--1082, 1972. \bibitem{MPT01} I.G. Macdonald, J.~Pach, and T.~Theobald. \newblock Common tangents to four unit balls in $\mathbb{R}^3$. \newblock {\em Discrete Comput. Geom.} 26:1--17, 2001. %\bibitem{pottmann-wallner-b2001} %H.~Pottmann and J.~Wallner. %\newblock {\em Computational Line Geometry}, %\newblock Springer-Verlag, Berlin, 2001. \bibitem{schubert-1886} H.~Schubert. Anzahlbestimmungen f\"ur lineare R\"aume beliebiger Dimension. {\em Acta Math.} 8:97-118, 1886. \bibitem{sedykh-shapiro-2002} V.~Sedykh and B.~Shapiro. Two conjectures on convex curves. Preprint, 2002. {\tt math.AG/0208218}~. \bibitem{So97a} F.~Sottile. \newblock Enumerative geometry for the real {G}rassmannian of lines in projective space. \newblock {\em Duke Math. J.} 87:59--85, 1997. \bibitem{So99a} F.~Sottile. \newblock The special {S}chubert calculus is real. \newblock {\em Electronic Research Announcements of the AMS} 5:35--39, 1999. \bibitem{M2} F.~Sottile. \newblock From enumerative geometry to solving systems of equations. \newblock In D.~Eisenbud, D.~Grayson, M.~Stillman, and B.~Sturmfels (eds.), {\em Computations in Algebraic Geometry with Macaulay $2$}, Algorithms and Computation in Mathematics, vol.~8, pp.~101--129, Springer-Verlag, 2001. %\bibitem{sottile-erag} %F.~Sottile. %\newblock Enumerative real algebraic geometry. %\newblock % In S.~Basu and L.~Gonzalez-Vega (eds.), % \emph{Algorithmic and Quantitative Real Algebraic Geometry}, % DIMACS series in Discrete Mathematics and Theoretical Computer Science, % vol.~60, AMS, pp. 139--180, 2003. \bibitem{STh02} F.~Sottile and T. Theobald. \newblock Lines tangent to $2n-2$ spheres in ${\mathbb{R}}^n$. \newblock \emph{Trans. Amer. Math. Soc.} 354:4815-4829, 2002. \bibitem{sturmfels-cbms} B.~Sturmfels. \newblock Solving Systems of Polynomial Equations. \newblock CBMS Regional Conference Series in Mathematics, vol.~97, AMS, Providence (RI), 2002. %\bibitem{Th02} %T.~Theobald. %\newblock An enumerative geometry framework for algorithmic line problems in % $\mathbb{R}^3$, %\newblock \emph{SIAM J.\ Comput.} 31:1212--1228, 2002. \end{thebibliography} %\bibliographystyle{amsplain} %\bibliography{bibl} \end{document}