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   \Blatt{13}
   \Abgabe{11/02/2002}

   Throughout this example sheet $K$ will be an algebraically closed field.


   \Nr
   Let $X\subset \A_{\C}^4$ be the surface given by the
   parametrization $\A_{\C}^2\rightarrow\A_{\C}^4:(s,t)\mapsto
   (s,st,t(t-1),t^2(t-1))$. 

   Find the embedding dimension of $X$ at
   $0$. Is $0$ regular or singular?

   {\scriptsize Hint, you may use Singular to find the generating ideal
     of $X$ resp.\ to verify your own result.}


   \Nr
   Show, a non-singular projective plane curve $C\subset\P_K^2$  is irreducible.

   \Nr\textbf{[Projective Jacobian Criterion]}\\[0.1cm]
     Let $X\subseteq \P_K^n$ be a 
     projective variety with
     $I(X)=(f_1,\ldots,f_r)\unlhd
     K\big[\;\underline{x}\;\big]$, and let $p=(p_0:\ldots:p_n)\in X$ be a
     point in $X$ with $d=\dim_p(X)$. 

     Show that  $p$ is regular if and
     only if $\rank\Big(\frac{\partial f_i}{\partial
       x_j}(p)\Big)_{i,j}=n-d$. 

     {\scriptsize Hint, pass to a suitable affine chart and use Euler's
       lemma, which says that if $f$ is a homogeneous polynomial of
       degree $d$, then $\sum x_i\frac{\partial f}{\partial
         x_i}=d\cdot f$. Note, you have to show that the above rank is 
       independant of the chosen coordinates of $p$!}

     \Nr
     Let  $X$ a quasi-projective variety 
     and $H$ a hypersurface in $X$ (i.\ e.\ a puredimensional subvariety of
     codimension 1). Let $p\in X$ such that $\ko_{X,p}$ is a unique
     factorization domain and that $p$ is a regular point of $H$.

     Show that $p$ is also a regular point of $X$. 

     {\scriptsize Hint, consider  the defining ideal $I_p\lhd \ko_{X,p}$ for $H$ in
       $p$ and show that $\height(I_p)$ is one.
       You may then use the following 
       result from commutative algebra without proof: \textit{A
         noetherian integral domain is a unique 
         factorization domain if and only if every prime ideal of
         height 1 is principal.}}


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