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\newcommand{\myTitle}{HW 12, due noon Thursday, May 16}
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\begin{enumerate}
  \item  Give geometric explanations for the following calculations in
    $A^*\G_1\proj^3$:
    \begin{enumerate}
      \item $(0,3)^2=(1,2)^2=(0,1)$;
	\medskip

	{\sc solution:}

      \item $(1,3)(0,3)=(1,3)(1,2)=(0,2)$;
	\medskip

	{\sc solution:}

      \item $(0,3)(1,2)=0$;
	\medskip

	{\sc solution:}

      \item $(1,3)^2=(0,3)+(1,2)$.
	\medskip

	{\sc solution:}

    \end{enumerate}
  \item The Chow ring of $\G_1\proj^4$.
    \begin{enumerate}
      \item List all the Schubert classes for $\G_1\proj^4$.  For each class,
	state the codimension, give both the $a$-notation and
	$\lambda$-notation, and describe the associated Schubert condition.  For
	example, one line of your list will be
	\begin{center}
	  \begin{tabular}{ccl}
	    codim&class&condition\\\hline
	    $3$&$(1,3)=\{2,1\}$&meet a line lying in a solid
	  \end{tabular}
	\end{center}
     a ``solid'' being a $3$-dimensional linear subspace.
	\medskip

	{\sc solution:}

  \item Make the $10\times10$ multiplication table for the Chow ring of
    $A^*(\G_1\proj^4)$.
	\medskip

	{\sc solution:}

  \item Find $m$ so that there will be a finite number of lines meeting $m$
    given lines, in general. Then find the number of lines meeting $m$ general
    lines.
	\medskip

	{\sc solution:}

  \item How many lines will meet $6$ planes, in general?
	\medskip

	{\sc solution:}

  \item What is the degree of $\G_1\proj^4\subset\proj^9$ (Pl\"ucker embedding)?
    (Hint:  The {\em degree} is the number of times a generic linear space of
    complementary dimension intersects the set. Since
    $\G_1\proj^4$ has dimension $(r+1)(n-r)=6$, i.e., codimension $3$,
    the degree is given by the number of times a general solid ($3$-dimensional subspace, codimension
    $6$) in $\proj^9$ meets the Grassmannian.  To create this solid, note that
    the Schubert class of codimension $1$ is given by intersecting the
    Grassmannian with a hyperplane.)
	\medskip

	{\sc solution:}

  \item Create your own enumerative problem that can be answered using this Chow
    ring.
	\medskip

	{\sc solution:}

  \end{enumerate}
  \item {\bf Linear duality.}  It turns out that taking the dual of the short
    exact sequence of vector spaces,
    \[
    0\to W\to V\to V/W\to0
    \]
    gives a short exact sequence:
    \[
    0\to(V/W)^*\to V^*\to W^*\to0.
    \]
   This gives rise to an isomorphism of Grassmannians:
   \begin{eqnarray*}
     G(r,V)&\to& G(n-r,V*)\\
     W&\to&(V/W)^*
   \end{eqnarray*}
   where $n=\dim V$. Fixing an inner product on $V$ identifies $V$ with $V^*$,
   and then the isomorphism of Grassmannians is given by
   sending $W$ to its orthogonal complement.
   Picking a basis for $V$, we get $G(r,n)\approx G(n-r,n)$.
   In projective language, 
   \[
   \G_r\proj^n=G(r+1,n+1)\approx G(n-r,n+1)=\G_{n-r-1}\proj^n.
   \]
   A special case is the duality between points and hyperplanes in $\proj^n$
   given by
   \[
   (a_0,\dots,a_n)\quad\leftrightarrow\quad a_0x_0+\dots+a_nx_n=0.
   \]
     There is a corresponding duality with the Chow rings
    for Grassmannians.  The dual for the Schubert class
    $\{\lambda_0,\dots,\lambda_r\}$ in the Chow ring for $\G_r\proj^n$ having
    Young diagram $Y_{\lambda}$ is the class in the Chow ring for
    $\G_{n-r-1}\proj^n$ having as its Young diagram the transpose of
    $Y_{\lambda}$.  For instance, the class $\{3,2\}\in A^5\G_1\proj^4$ is
    dual to $\{2,2,1\}\in A^5\G_2\proj^4$:
    \[
    \Yvcentermath1
    \yng(3,2)\quad\leadsto\quad\yng(2,2,1)
    \]
    Each of the enumerative problems we have considered using the Schubert
    calculus has an associated dual problem.  For instance, in the plane, there
    is one line containing two given points and there is one point contained in
    two lines.  
    \begin{enumerate}    
  \item List all the Schubert classes for $\G_2\proj^4$, their codimensions,
    and the relevant Schubert conditions.  Match each class with its dual class
    in the Chow ring for $\G_1\proj^4$.  (Compare the Schubert conditions for
    each class and its dual.  Note how the word ``contains'' is dual to
    ``contained in'' but the word ``meeting'' is self-dual.)
	\medskip

	{\sc solution:}

  \item What are the duals of the enumerative problems in 2 (c), (d), and (f)?
	\medskip

	{\sc solution:}

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