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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Winterterm 1997/98 \\
   Mathematical Department \hfill 23rd November, 1997 \\
   Gert-Martin Greuel \hfill Thomas Keilen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Introduction to Algebraic Geometry \\
   \end{center}
   \vspace{0.5cm}
   \normalsize
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   \textbf{Due date:} 1st December, 1997
   \\[0.2cm]
   
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   \setcounter{exercise}{16}

   \begin{exercise}
     Show that the Veronese embedding $\rho_d:\P_K^n\rightarrow\P_K^N$
     of exercise 15 is an isomorphism of $P^n_K$ to its image.
   \end{exercise}

   \begin{exercise}
     Let $K$ be any field, and let $H$ be an irreducible
     hypersurface in $\P_K^n$.
     Show that $\P^n_K\backslash H$ is isomorphic to an affine
     algebraic set. (In particular, every projective algebraic set
     over $\R$ is affine! - Compare this with exercise 20.)\\
     \small (Hint, if $H$ has degree $d$, i.\ e.\ if $H=V(f)$ with $f$
     irreducible and $\deg(f)=d$, then use the Veronese embedding
     $\rho_d$.) 
   \end{exercise}

   \begin{exercise}
     Let $K$ be an algebraically closed (!) field. Show that every irreducible conic
     in $\P^2_K$, i.\ e.\ every irreducible plane curve of degree 2, is
     isomorphic to $\P_K^1$.\\
     (Hint, show every two irreducible conics are isomorphic by
     reducing the equations to some normal form, and then use a
     Veronese embedding.) 
   \end{exercise}

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