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%   \Abgabe{12/11/2001}

   Let $K$ be any field.

   \Nr\textbf{[Twisted Cubic]}\\[0.1cm]
   Is the set $V=\big\{\big(t,t^2,t^3\big)\in
   K^3\;\big|\;t\in K\big\}$ an affine algebraic subset of $\A^3_K$?
   If yes, what is the vanishing ideal $I(V)$ of $V$? Show that $V$ is
   homeomorphic to $\A_K^1$. Show that any three different points
   of $V\setminus\{(0,0,0)\}$ are linearly independend as vectors in $K^3$.\footnote{We
     say, the points on $V$ are in general position.}

   \Nr
   Let
   $V=V\big(x^2-yz,xz-x\big)\subset\A_K^3$. Show that $V=V(P_1)\cup
   V(P_2)\cup V(P_3)$ for suitable
   $P_1,P_2,P_3\in\Spec\big(K[x,y,z]\big)$.\footnote{For a ring $R$ we
     set $\Spec(R)=\{P\unlhd R\;|\;P\text{ prime}\}$.}

   \Nr\textbf{[Tangent Cone]}\\[0.1cm]
   Let $0\not=f\in K[x,y]$ and $C=V(f)\subset\A_K^2$. How could one
   define the tangent line(s) at the curve $C$ in the origin $0=(0,0)$? 
   Consider the examples $f=x$, $f=y-x^2$, $f=x^2-y^3$, $f=x^2-y^2$,
   and $f=y^2-x^4$.

   \Nr\textbf{[Neil's Parabola and Newton's Node]}
   \begin{enumerate}
   \item Let $V=V\big(y^2-x^3\big)$. Show that $K[V]:=K[x,y]/I(V)$ is
     isomorphic to $K\big[t^2,t^3\big]$.
   \item    Let $V=V\big(y^2-x^2-x^3\big)\subset\A_K^2$. Show that there is
     rational parametrisation $\varphi:A_K^1\rightarrow
     \A_K^2:t\mapsto(f,g)$ with $f,g\in K[t]$ and
     $\varphi\big(\A_K^1\big)=V$. 
   \end{enumerate}
   
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