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\newcommand{\myTitle}{HW 11, due Friday, April 26}
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\begin{enumerate}
  \item A {\em linear subspace $L$ of $\proj^n$ of dimension $r$},
    also called an {\em $r$-plane}, is an $(r+1)$-dimensional vector subspace
    $\tilde{L}\subseteq\A^{n+1}$ modulo the equivalence $p\sim\lambda p$ for
    every $p\in \tilde{L}$ and nonzero $\lambda\in k$. Given two linear subspaces $L$ and $M$ of
    $\proj^n$, define their {\em span} to be the linear subspace determined by
    $\tilde{L}+\tilde{M}$ in $\A^{n+1}$.  In other words, a point of
    $\mbox{Span}\{L,M\}\subseteq\proj^n$ is
    any nonzero $p+q$ with $p\in\tilde{L}$ and $q\in\tilde{M}$, modulo
    scaling. The intersection of linear spaces in $\proj^n$ is a linear space,
    possibly empty (corresponding to the linear subspace
    $\{0\}\subset\A^{n+1}$). Note that $\dim L=\dim\tilde{L}-1$. 
    \begin{enumerate}
      \item Show that
	\[
	\dim\,\mbox{Span}\{L,M\}=\dim L+\dim M-\dim(L\cap M)
	\]
	where the dimension of the empty set is taken to be $0$.  (Please use a
	nice short exact sequence to prove this.)
	\medskip
	
	{\sc solution:}

      \item Show that $\dim (L\cap M)\geq \dim
	L+\dim M-n$.  In particular, $L\cap M\neq\emptyset$ if $\dim
	L+\dim M\geq n$.
	\medskip
	
	{\sc solution:}

      \item Thus, two planes, i.e., two $2$-planes, in $\proj^4$ must meet in a linear
	space of dimension at least $0$, i.e., at least in a point.  Give
	explicit examples showing that two planes in $\proj^4$ can meet in a
	point, a line, or a plane.  (There is more room in
	$4$-space than in $3$-space, where two planes must meet in at least a
	line.)
	\medskip
	
	{\sc solution:}

    \end{enumerate}
  \item For each flag of linear spaces
\[
A_0\subsetneq \dots\subsetneq A_r
\]
we defined the Schubert variety
\[
  \mathfrak{S}(A_0,\dots,A_r)=\{L\in\G_r\proj^n:\dim(L\cap A_i)\geq i\mbox{ for
  all $i$}\}
\]
and the corresponding Schubert class $(a_0,\dots,a_r)\in A^*(\G_r\proj^n)$
where $a_i=\dim A_i$.  (Working modulo rational equivalence of cycles, only the
dimensions of the $A_i$ matter.)  Each Schubert class represents $r$-planes in
$\G_r\proj^n$ satisfying certain conditions.  For instance, in
$A^*(\G_1\proj^3)$, the class $(0,2)$ is the class of a Schubert variety
$\mathfrak{S}(A_0,A_1)$ where $A_0$ is a point and $A_1$ is a plane.  This
Schubert variety consists of lines $L$ passing through the point $A_0$ and
lying in the plane $A_1$ (in order to satisfy the conditions $\dim(L\cap
A_i)\geq i$).

\begin{enumerate}
  \item  Let $n=3$.  For $r=0,1,2,3$, describe all possible Schubert classes
    $(a_0,\dots,a_r)$ and state the corresponding condition placed on
    $r$-planes in $\proj^3$. (Note that $0\leq a_0<\dots<a_r\leq 3$.)
	\medskip
	
	{\sc solution:}

  \item The class $(a_0,\dots,a_r)$ is an element of $A^{\ell}(\G_r\proj^n)$ where the
    codimension $\ell$ is given by
    \begin{displaymath}
      \ell=(r+1)(n-r)-\sum_{i=0}^r(a_i-i).
    \end{displaymath}
    Calculate the codimension for each of the classes in part (a).
	\medskip
	
	{\sc solution:}

\end{enumerate}
\item[3.] The set of lines in $\proj^3$ meeting a given line $L$ is a
  hyperplane section of $\G_1\proj^3\subset\proj^5$, where as usual we
  consider the Grassmannian embedded via the Pl\"ucker embedding.  In other
  words, there is a hyperplane $H_L\subset\proj^5$ such that the lines
  meeting $L$ are given by $\G_1\proj^3\cap H_L$.  The collection of lines
  meeting $4$ given lines $L_1,\dots,L_4$ is then 
  \[
  \cap_i(\G_1\proj^3\cap H_{L_i})=\G_1\proj^3\cap\left( \cap_iH_{L_i} \right).
  \]
  If the $L_i$ are general lines, then $\cap_iH_{L_i}$ will be a line in
  $\proj^5$, which we expect to meet the quadric hypersurface
  $\G_1\proj^3\subset\proj^5$ in two points.  The point of this exercise is to
  compute an explicit example.

  The first step is to get explicit equations for the hyperplanes described
  above.
  A line~$L$ in $\proj^3$ can be parametrized by $L(s,t)=sp+tq$ where
  $p,q\in\proj^3$ are any two fixed distinct points on the line and $(s,t)$
  varies over $\proj^1$.  Consider the matrix
  \[
  C=
  \left(\begin{array}{cccc}
    p_0&p_1&p_2&p_3\\
    q_0&q_1&q_2&q_3\\
    x_0&x_1&x_2&x_3\\
    y_0&y_1&y_2&y_3\\
  \end{array}\right).
  \]
  The first two rows give the homogeneous coordinates of $L$ as an element of
  the Grassmannian: their span is $L$.  The last two rows span an
  arbitrary line, $M$, in $\proj^3$.  The lines $L$ and $M$  intersect in
  $\proj^3$ if and only if their corresponding $2$-dimensional subspaces in
  $\A^4$ meet in a subspace of at least affine dimension $1$.  In other words,
  if and only if $C$ has a non-trivial kernel.  So the condition
  that $L\cap M\neq\emptyset$ is equivalent to $\det C=0$.  To get the equation
  of the hyperplane, calculate $\det C$ using a generalized Laplace expansion
  along the first two rows of $C$.
\medskip

\noindent{\bf Generalized Laplace Expansion.}  Let $D$ be an $n\times n$ matrix.
Let $[n]=\{1,\dots,n\}$, and fix row indices $I\subset[n]$.  The complement is
denoted $I^c=[n]\setminus I$.  For each collection of column indices $J$ having
the same number of elements as $I$, i.e., $|I|=|J|$, we write $D_{I,J}$ for the corresponding
submatrix of $D$.  Then
\[
\det D=\sum_{J\subset[n],|J|=|I|}(-1)^{\sum_{i\in I}i+\sum_{j\in J}j}\det D_{I,J}\det
D_{I^c,J^c}.
\]
 
Expanding $\det C$ along the first two rows gives
\begin{eqnarray*}
  \det C&=& (p_0q_1-p_1q_0)(x_2y_3-x_3y_2)-(p_0q_2-p_2q_0)(x_1y_3-x_3y_1)\\
  &&\quad+(p_0q_3-p_3q_0)(x_1y_2-x_2y_1)+(p_1q_2-p_2q_1)(x_0y_3-x_3y_0)\\
 &&\quad-(p_1q_3-p_3q_1)(x_0y_2-x_2y_0)+(p_2q_3-p_3q_2)(x_0y_1-x_1y_0).
\end{eqnarray*}
Note that the Pl\"ucker coordinates for the line spanned by $x$ and $y$ are
\[
(x_0y_1-x_1y_0,x_0y_2-x_2y_0,x_0y_3-x_3y_0,x_1y_2-x_2y_1,x_1y_3-x_3y_1,x_2y_3-x_3y_2).
\]
Let $z_{01},z_{02},z_{03},z_{12},z_{13},z_{23}$ be the coordinates on $\proj^5$, and let 
\[
[ij]=\det
\left(\begin{array}{cc}
  p_i&p_j\\
  q_i&q_j
\end{array}\right).
\]
We see that $\det C=0$ defines
the intersection of $\G_i\proj^3\subset\proj^5$ with the hyperplane
\[
H_L=[23]z_{01}-[13]z_{02}+[12]z_{03}+[03]z_{12}-[02]z_{13}+[01]z_{23}=0.
\]
    Let $(u,x,y,z)$ be the coordinates on $\proj^3$, and consider the four lines
    given in homogeneous equations by
      \[
      \begin{array}{ll}
      L_1=\{y=z=0\},&\quad L_2=\{x=z,y=u\}\\
      L_3=\{x=2z,y=2u\},&\quad L_4=\{x=y,z=u\}.
    \end{array}
    \]
      \begin{enumerate}
	\item Compute the four hyperplanes, $H_{L_i}$.
	\medskip
	
	{\sc solution:}

	\item Compute a parametric equation for the line
	  $\cap_{i=1}^4H_{L_i}$.
	\medskip
	
	{\sc solution:}

	\item Find the two points of intersection of that line with the
	  Grassmannian.
	\medskip
	
	{\sc solution:}

	\item Describe these two points as lines in $\proj^3$.
	\medskip
	
	{\sc solution:}

      \end{enumerate}
\end{enumerate}
\end{document}
