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\newcommand{\myTitle}{HW 10, due Friday, April 19}
\newcommand{\myClass}{Math 411}
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\begin{enumerate}
 \item
Let $P$ be the hexagon in the plane with vertices $(0,0)$, $(1,0)$,
$(0,1)$, $(2,1)$, $(1,2)$, and $(2,2)$.
\begin{enumerate}
\item  Draw the fan, $\Delta$, for the corresponding toric variety,
  $X$, labeling the
  first lattice points along its $1$-dimensional cones.
  \medskip

  {\sc solution:}

\item Choose two adjacent $2$-dimensional cones in $\Delta$,
  construct the two corresponding affine toric varieties, and show the
  gluing instructions.
  \medskip

  {\sc solution:}

\item Show that $X$ smooth.
  \medskip

  {\sc solution:}

\item Calculate the Chow ring, $A^{\bullet}(X)$.
  \medskip

  {\sc solution:}

\item Calculate the cohomology $H^kX$ for all $k$.
  \medskip

  {\sc solution:}

\item Give the mapping of $X$ into $\proj^6$ determined by the $7$
  lattice points of $P$ (there is one interior point) in homogeneous
  coordinates.
  \medskip

  {\sc solution:}

\item Pick two components of your mapping, and show that they have the
  same degree.
  \medskip

  {\sc solution:}

\item Describe $X$ as a quotient, $(\C^{\Delta(1)}\setminus Z)/(x\sim\ell\cdot
  x)$
  according to Cox's theorem.
  \medskip

  {\sc solution:}

\end{enumerate} 
\item  We have seen that the toric variety $Y$ determined by the single cone,
  $\sigma=\R_{>0}(2,-1)+\R_{>0}(0,1)$, has semigroup algebra 
  \[
  \C[x,xy,xy^2]\approx \C[u,v,w]/(uw-v^2).
  \]
  Thus, $Y=\{(u,v,w)\in\C^3: uw=v^2\}$, a cone.  The fact that $Y$ has a
  singularity can be inferred from the cone since
  \[
  \det\left(\begin{array}{rr}
    -2&1\\
    0&1
  \end{array} \right)
  = -2\neq \pm1.
  \]
  To desingularize $Y$ we ``blow-up'' the singular point by splitting
  $\sigma$ into two ``nonsingular'' cones.  Let $\Delta$ be the fan with maximal
  cones
  \begin{eqnarray*}
    \sigma_1 &=& \R_{>0}(2,-1)+\R_{>0}(1,0)\\
    \sigma_2 &=& \R_{>0}(1,0)+\R_{>0}(0,1),
  \end{eqnarray*}
  and let $X=X(\Delta)$ be the corresponding toric variety.
  \begin{enumerate}
    \item Describe how $X$ is obtained from gluing two copies of $\C^2$.
  \medskip

  {\sc solution:}

    \item Describe a mapping $\pi\colon X\to Y$ such that $\pi^{-1}(p)$ consists
      of a single point for all $p\in Y\setminus\{(0,0,0)\}$ and such that
      $\pi^{-1}(0,0,0)$ is a ``line''.  (Letting $x$ and $y$ be the
      indeterminates corresponding to the lattice points $(1,0)$ and $(0,1)$,
      respectively, and writing all coordinate functions in terms of $x$ and $y$
      should guide the way. Show how the mapping is defined on the two copies of
      $\C^2$ that are glued to form~$X$.)
  \medskip

  {\sc solution:}

  \end{enumerate}

\item 
  \begin{enumerate}
    \item Suppose $L\colon V\to W$ and $M\colon V'\to W'$ are linear mappings of
  finite-dimensional vector spaces.  There is an induced mapping
  \begin{eqnarray*}
    L\otimes M\colon V\otimes V'&\to& W\otimes W'\\
    v\otimes v'&\mapsto& L(v)\otimes M(v')
  \end{eqnarray*}
  Choosing bases $v_1,\dots, v_n$ for $V$; $w_1,\dots,w_m$ for $W$;
  $v_1',\dots,v_t'$ for $V'$; and $w_1',\dots,w_s'$ for $W'$, we identify $L$
  and $M$ with matrices.  Choosing the corresponding bases
  \[
v_1\otimes v_1',v_1\otimes v_2',\dots,v_1\otimes v_t',v_2\otimes
v_1',\dots,\dots,v_n\otimes v_t'
  \]
  for $V\otimes V'$ and 
\[
w_1\otimes w_1',w_1\otimes w_2',\dots,w_1\otimes w_s',w_2\otimes
w_1',\dots,\dots,w_m\otimes w_s'
\]
   for $W\otimes W'$ (in the given orders) describe 
  $L\otimes M$ as a matrix.
  \medskip

  {\sc solution:}

  \item  With the above conventions, what is $L\otimes M$ for
    \[
    L=
    \left(\begin{array}{ccc}
      1&2&3\\
      4&5&6
    \end{array} \right),
    \qquad
    M=
    \left(\begin{array}{cc}
      1&2\\
      3&4
    \end{array} \right).
    \]
  {\sc solution:}

  \end{enumerate}
\item Consider an exact sequence of finite-dimensional vectors spaces:
  \[
0\to V_1\to V_2\to\dots\to V_k\to0.
  \]
  Prove that $\sum_{i=1}^k (-1)^i\dim V_i=0$.
  \medskip

  {\sc solution:}

\end{enumerate}
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