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% Author Package file for use with AMS-LaTeX 1.2
\controldates{26-MAR-1999,26-MAR-1999,26-MAR-1999,26-MAR-1999}
 
\documentclass{era-l}
\usepackage{amssymb}
\usepackage{epsf}


\newtheorem{thm}{Theorem}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{conj}[thm]{Conjecture}

\theoremstyle{remark}
\newtheorem*{remark}{Remark}
\newtheorem*{claim}{Claim}

\begin{document}

\title[THE SPECIAL SCHUBERT CALCULUS IS REAL]
{The special Schubert calculus is real}   

\author{Frank Sottile}
\address{Mathematical Sciences Research Institute,
        1000 Centennial Drive,
        Berkeley, CA 94720}
\curraddr{Department of Mathematics,
University of Wisconsin,
         Van Vleck Hall,
         480 Lincoln Drive,
         Madison, Wisconsin 53706-1388}
\urladdr{http://www.math.wisc.edu/\~{}sottile}
\email{sottile@math.wisc.edu}


\issueinfo{5}{05}{}{1999}
\dateposted{April 1, 1999}
\pagespan{35}{39}
\PII{S 1079-6762(99)00058-X}
\def\copyrightyear{1999}
\copyrightinfo{1999}{American Mathematical Society}

\subjclass{Primary 14P99, 14N10, 14M15, 14Q20; Secondary 93B55}
\date{December 20, 1998}

\commby{Robert Lazarsfeld}

\keywords{Schubert calculus, enumerative geometry, Grassmannian, pole
placement problem} 

\thanks{MSRI preprint \# 1998-067.}
\thanks{Research supported by NSF grant DMS-9701755.}

\begin{abstract}
We show that  the Schubert calculus of enumerative geometry is real, for
special Schubert conditions.
That is, for any such enumerative problem, there exist real conditions for
which all the \textit{a priori} complex solutions are real.
\end{abstract}


\maketitle

Fulton asked how many solutions to a problem of enumerative
geometry can be real, when that problem is one of counting geometric figures
of some kind having specified position with respect to some general fixed
figures~\cite{Fu_84}.
For the problem of plane conics tangent to five general conics, the
(surprising) answer is that all 3264 may be real~\cite{RTV}. 
Recently, Dietmaier has shown that all 40 positions of the Stewart
platform in robotics may be real~\cite{Dietmaier}.
Similarly, given any problem of enumerating lines in projective space
incident on some general fixed linear subspaces, there are real fixed
subspaces such that each of the (finitely many) incident lines are
real~\cite{Sottile97a}. 
Other examples are shown in~\cite{Sottile97c,Sottile97b},
and the case of 462 4-planes meeting 12 general 3-planes in ${\mathbb R}^7$
is due to an heroic symbolic computation~\cite{FRZ}.

For any problem of enumerating $p$-planes having excess  intersection with a
collection of fixed planes, we show there is a choice of fixed 
planes osculating a rational normal curve at real points so that each of the
resulting $p$-planes is real. 
This has implications for the problem of placing real poles in linear systems
theory~\cite{Byrnes} and is a special case of a far-reaching conjecture of
Shapiro and Shapiro~\cite{Sottile_shapiro}.

\section*{Special Schubert conditions}

For background on the Grassmannian, Schubert cycles, and the Schubert
calculus, see any of~\cite{Hodge_Pedoe,Griffiths_Harris,Fulton_tableaux}.
Let  $m,p\geq 1$ be integers.
Let $\gamma$ be a rational normal curve in ${\mathbb R}^{m+p}$.
For $k>0$ and $s\in\gamma$, let $K_k(s)$ be the $k$-plane
osculating $\gamma$ at $s$.
For every integer $a>0$, let $\tau_a(s)$ be the special Schubert cycle
consisting of $p$-planes $H$ in ${\mathbb C}^{m+p}$ which meet 
$K_{m+1-a}(s)$ nontrivially, and let
$\tau^a(s)$ be the special Schubert cycle consisting of 
$p$-planes $H$ in ${\mathbb C}^{m+p}$ meeting $K_{m-1+a}(s)$
improperly: 
$\dim H\cap K_{m-1+a}(s)> a-1$.
These cycles $\tau_a(s)$ and $\tau^a(s)$ each have
codimension $a$ and $\tau^1=\tau_1$. 
Recall that the Grassmannian of $p$-planes in ${\mathbb C}^{m+p}$ has
dimension $mp$.
For any Schubert condition $w$, let $\sigma_w(s)$ be the Schubert cycle of
type $w$ given by the flag osculating $\gamma$ at $s$
and set $|w|$ to be the codimension of $\sigma_w(s)$.


\begin{thm}\label{thm:1}
Let $a_1,\ldots,a_n$ be positive integers with $a_1+\cdots+a_n=mp$.
For each $i=1,\ldots,n$ let $\sigma_i(s)$ be either $\tau^{a_i}(s)$ or
$\tau_{a_i}(s)$.
Then there exist real points $0,\infty,s_1,\ldots,s_n\in{\gamma}$ such that
for any Schubert conditions $w,v$, and integer $k$ with 
$|w|+|v|+a_k+\cdots+a_n=mp$, the intersection
\begin{equation}\label{eq:special}
  \sigma_w(0)\cap \sigma_v(\infty)\cap
  \sigma_k(s_k)\cap\cdots\cap\sigma_n(s_n)
\end{equation}
is transverse with all points of intersection real.
\end{thm}

Our proof is inspired by the Pieri homotopy algorithm of~\cite{HSS}.

We prove this in the case that each
$\sigma_i(s)=\tau_{a_i}(s)$.
This is no loss of generality, as the cycles $\tau_a(s)$ and $\tau^a(s)$
share the properties we need.
We use the following two results of Eisenbud and Harris~\cite{EH83},
who studied such intersections in their theory of limit linear systems.
For a Schubert class $w$, let
$w*a$ be the index of summation in the Pieri formula in the
cohomology of the Grassmannian~\cite{Fulton_tableaux},
\begin{equation*}
\sigma_w\cdot \tau_a\ =\  \sum_{v\in w*a} \sigma_v .
\end{equation*}

\begin{prop}\label{prop:only}
\mbox{ }
\begin{enumerate}
\item (Theorem 2.3 of~\cite{EH83}) Let $s_1,\ldots,s_n$ be distinct points on
$\gamma$  and $w_1,\ldots,w_n$ be Schubert
conditions.
Then the intersection of Schubert cycles
\begin{equation*}
\sigma_{w_1}(s_1)\cap\sigma_{w_2}(s_2)\cap\cdots\cap\sigma_{w_n}(s_n)
\end{equation*}
is proper in that it has dimension $mp-|w_1|-\cdots-|w_n|$.

\item (Theorem 8.1 of~\cite{EH83}) For any Schubert condition $w$,
integer $a>0$, and $0\in\gamma$,
we have 
\begin{equation*}
\lim_{t\rightarrow 0}\left( \sigma_w(0)\cap\tau_a(t)\rule{0pt}{11pt}\right)\
=\  \bigcup_{v\in w*a} \sigma_v(0),
\end{equation*}
the limit taken along the rational normal curve, and as schemes.
\end{enumerate}
\end{prop}

\begin{proof}[Proof of Theorem~1]
We argue by downward induction on $k$.
The initial case of $k=n$ holds as Pieri's formula implies the intersection
is a single, necessarily real, point.
Suppose it holds for $k$, and let $w,v$ satisfy 
$|w|+|v|+a_{k-1}+\cdots+a_n=mp$.

\begin{claim}
The cycle
\begin{equation*}
\sum_{u\in w*a_{k-1}}
\sigma_u(0)\cap\sigma_v(\infty)\cap
\tau_{a_k}(s_k)\cap\cdots\cap\tau_{a_n}(s_n)
\end{equation*}
is free of multiplicities.

If not, then two summands, say $u$ and $u'$, have a point in common
and so
\begin{equation}\label{eq:bad_intersect}
  \sigma_u(0)\cap\sigma_{u'}(0)\cap\sigma_v(\infty)\cap
  \tau_{a_k}(s_k)\cap\cdots\cap\tau_{a_n}(s_n)
\end{equation}
is nonempty.
However, $\sigma_u(0)\cap\sigma_{u'}(0)$
is a Schubert cycle of smaller dimension.
Thus the intersection~(\ref{eq:bad_intersect}) 
must be empty, by Proposition~\ref{prop:only}~(1).


From the claim and Proposition~\ref{prop:only}~(2), there is an
$\epsilon_{w,v}>0$ 
such that if $0