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\begin{document}
\pagestyle{fancy} \lhead{GIT, Assignment 2}
\rhead{Due date: March 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
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\begin{exercise}
Read section 1.3 and, after finishing the assignment, section 1.4 in Schmitt.
\end{exercise}

\begin{exercise}
  Let $G=\CC^*$ act on $V=\CC^3$ via $\lambda\mapsto
  \diag(\lambda^{-a},\lambda^b,\lambda^c)\in\GL_3(\CC)$, where $a,b,c\in\ZZ_{>0}$.
\begin{enumerate}
\item Describe the invariant ring $\CC[V\sslash G]$ for $a=b=1$ and
  $c=2$, and describe the categorical quotient variety.
\item Under what conditions on $a,b,c$ is the invariant ring a polynomial ring?
\end{enumerate}
\end{exercise}

\begin{exercise}
  Let $G$ be a reductive linear algebraic group and let $K\subset G$
  be a compact real form.  If $G$ is acting on a finite-dimensional
  vector space $V$, with action map $\rho:K\times V\to V$, prove that 
\[
f\mapsto \int_K (\rho^* f) dv_K
\]
defines a Reynolds operator $\CC[V]\to\CC[V]^G$. Try also to prove it without using Zariski density -- use the polar decomposition of $G$. 
\end{exercise}

\begin{exercise}
  Suppose that $G$ is affine algebraic group acting on affine variety
  $V$. Suppose that the categorical quotient $V\sslash G$ exists. Show that
  $\CC[V]^G$ must be finitely generated.
\end{exercise}

\begin{exercise}
  Suppose that $G$ is a reductive affine algebraic group acting on an
  affine variety $V$ with categorical quotient $\pi:V\to V\sslash G$. Show
  that if $W\subset V$ is a closed $G$-invariant subvariety, then
  $\pi(W)$ is a closed subvariety, and defines the categorical quotient of the
induced $G$-action on $W$.
\end{exercise}

\begin{exercise}
  Let $G$ be an affine algebraic group acting on an affine algebraic variety $V$.  Prove that each orbit $\OO_v$, $v\in V$, is open in the closed variety $\overline{\OO_v}\subset V$.
\end{exercise}


\begin{exercise}
  Let $GL_n(\CC)$ act on $\End(\CC^n)$ by the adjoint action.  Prove
  that the ring of invariants is a polynomial ring generated by the
  coefficients of the characteristic polynomial.  Prove that an alternative generating
set consists of the functions given by 
\[
\End(\CC^n)\ni A\mapsto \Tr^k(A) \in \CC. 
\]
Describe the closed orbits parametrized by this categorical quotient. 
\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. 

