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\begin{document}
\pagestyle{fancy} \lhead{GIT, Assignment 1}
\rhead{Due date: February 14, 2014} \cfoot{} 

\noindent{\it Part of the assignment is to understand the language used in
the statement of the problems: work together and ask me questions
after class.  With the exception of the bonus question, I am not
asking for any hard labour; you can give minimalistic arguments.}

\begin{exercise}
  $\CC^*\times\CC^*$ acts on $\CC^2$ via $(\lambda_1,\lambda_2)\cdot
  (x,y) = (\lambda_1 x, \lambda_2 y)$. We obtain $\CC P^1$ by taking
  the quotient of $\CC^2\setminus\{0\}$ by the diagonal subgroup
  $\CC^*$. The residual $\CC^*$ action on $\CC P^1$ is then given by
  \begin{equation}
    \label{eq:1}
    \lambda\cdot [x:y]= [\lambda x: y] = [x:\lambda^{-1}y].
  \end{equation}
  Describe the orbits of the action, as well as the relation of
  containment of orbit closures.  Contrast this with the orbits for
  the induced action by the compact group $U(1)\subset \CC^*$.  
  
  Now consider the $\OO(1)$ line bundle; recall that it can be
  described as a quotient of $(\CC^2\setminus\{0\})\times \CC$ by the
  $\CC^*$ action 
  \begin{equation}
    \label{eq:2}
    \mu\cdot ((x,y),z) = ((\mu x, \mu y), \mu z).
  \end{equation}
  
\begin{itemize}
\item[a)] Extend the $\CC^*$ action on $\CC P^1$ to an action on the total
  space of the $\OO(1)$ line bundle, in such a way that the action is
  by vector bndle morphisms.  Describe any choices involved.
\item[b)] Describe the orbit structure and orbit closure relations for
  $\OO(1)$ and how these depend on the possible choices above.
\end{itemize}
  
\end{exercise}

\begin{exercise}
  As in class, define the line bundle $\OO(n)$ over $\PP(V)$ by
  exhibiting its total space as a quotient of
  $V\setminus\{0\}\times\CC$ by the $\CC^*$-action
  \begin{equation}
    \label{eq:8}
    \lambda\cdot (v,z) = (\lambda v, \lambda^n z).
  \end{equation}
  Give trivializations of $\OO(n)$ over the affine charts given by the
  nonvanishing sets of a basis of sections of $\OO(1)$, and compute
  transition functions for the line bundle $\OO(n)$.  Prove that
  $\OO(m)\otimes\OO(n) = \OO(m+n)$ as line bundles.
\end{exercise}


\begin{exercise}
  Give an argument explaining the fact that the tangent space to
  $\PP(V)$ at the point $[v]$, $v\in V\setminus\{0\}$ can be naturally
  identified with the space of linear maps from $[v]$ to
  $V/[v]$. Using the Euler exact sequence which defined the
  tautological line bundle, namely
  \begin{equation}
    \label{eq:7}
    \xymatrix{0\ar[r]& \OO(-1)\ar[r] & \underline{V}\ar[r] & Q\ar[r] & 0},
  \end{equation}
  show that the tangent bundle to $\CC P^1$ is isomorphic to $\OO(2)$.  Show furthermore that the determinant line bundle of $T\PP(V)$ is in general $\OO(\dim V)$.
\end{exercise}


\begin{exercise}
  The group $PGL(2,\CC)$ is the quotient of $GL(2,\CC)$ by its center.
  Show that $PGL(2)$ is an affine variety.  
\end{exercise}



\begin{exercise}
  Prove that the homogeneous ideal cutting out the pair of points
  \{[1:0:0],[0:1:0]\} in $\CC P^2$ is $(X_2, X_0X_1)$. Describe
  geometrically why this is the case. 
\end{exercise}

\begin{exercise}
  Let $U=\CC^{n+1}$ and $V = \CC^{m+1}$. The projective space
  $\PP(\Hom(U,V))$ of projective matrices has various natural subsets
  such as the set of matrices of rank $\leq k$. These subsets are cut
  out by homogeneous ideals generated by the minors of size $k+1$.
\begin{itemize}
\item[a)] Show that in the case $n=m=1$, the rank $\leq 1$ locus is a
  smooth projective quadric in 3 dimensions.
\item[b)] Show that in this case the quadric may be identified with
  $\PP^1\times\PP^1$ via the map $\PP(U^*)\times \PP(V)\to
  \PP(\Hom(U,V))$ defined by
  \begin{equation}
    \label{eq:3}
    ([\xi],[v])\mapsto [\xi\otimes v].
  \end{equation}
\item[c)] Show that the curves $\PP^1\times\{q\}$ and
  $\{p\}\times\PP^1$ in the domain are mapped to projective lines in
  the codomain, defining two rulings of the quadric by lines. 
\end{itemize}  
  This is a special case of the \emph{Segre embedding} of
  $\PP(U)\times \PP(V)$ in $\PP(U\otimes V)$ via 
  \begin{equation}
    \label{eq:4}
    ([u],[v])\mapsto [u\otimes v].
  \end{equation}
\begin{itemize}
\item[d)] Write the Segre embedding in terms of homogeneous
  coordinates on the domain and codomain.
\item[e)] Describe a homogeneous ideal of the codomain which cuts out
  the image of the Segre embedding, known as the \emph{Segre variety}.
\item[f)] Explain which line bundle and linear system on the domain
  effectuates the Segre embedding.
\end{itemize}
\end{exercise}

\begin{exercise}
  The \emph{Veronese embedding} of degree $d$ is the map $\PP(V)\to
  \PP(\Sym^d(V))$ defined by 
  \begin{equation}
    \label{eq:5}
    [v]\mapsto[v^d = v\otimes\cdots\otimes v].
  \end{equation}
  The image is called the \emph{Veronese variety}.  In the case
  $\PP(V)=\CC P^1$, the Veronese variety is called the rational normal
  curve of degree $d$.
  \begin{itemize}
  \item[a)] In coordinates $[t_0:t_1]\in\PP^1$ and
    $[X_0:\ldots:X_d]\in \PP^d$, describe explicitly the embedding of $\PP^1$ as
    the rational normal curve.  
  \item[b)] Show the curve has degree $d$ by counting its intersection
    points with a generic projective hyperplane.
  \item[c)] Which linear system on $\CC P^1$ effectuates the embedding
    as a rational normal curve?  
  \end{itemize}
  The rational normal curve of degree $2$ is a smooth conic in
  $\PP^2$. In the degree $3$ case, it is called a \emph{twisted cubic}
  curve in $\PP^3$.  

  \noindent {\bf Bonus:} Show that the homogeneous ideal corresponding to
  the twisted cubic is the one defined by the $2\times 2$ minors of the matrix 
  \begin{equation}
    \label{eq:6}
    \begin{pmatrix}X_0 & X_1 & X_2\\X_1 & X_2 & X_3\end{pmatrix}
  \end{equation}
  To show that these three generators of degree 2 are contained in the
  ideal is easy; it is more difficult to show that any homogeneous
  polynomial $F$ in the ideal must be a combination of the above three generators. 
\end{exercise}




\end{document}
Line bundles CP1 - show tangent bundle is O(2)

Grassmannian - prove that det T on G(k,n) is O(-n-1) from Plucker

G(2,4) ruling by a and b planes




Quadrics
Pencil of curves
Twisted Cubic
Veronese

describe a graded algebra corresponding to an elliptic curve. 

