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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Winterterm 1997/98 \\
   Mathematical Department \hfill 08th February, 1998 \\
   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:} 16th February, 1998
   \\[0.2cm]
   
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   \setcounter{exercise}{46}


   \begin{exercise}[Projective Jacobian Criterion]
     Let $K$ be algebraically closed, and let $X\subseteq \P_K^n$ be a 
     projective variety with
     $I(X)=(f_1,\ldots,f_r)\unlhd
     K[\underline{X}]$. $P=(p_0:\ldots:p_n)\in X$ be a point in $X$
     with coordinates
     $\underline{p}=(p_0,\ldots,p_n))$ and with $d=\dim_P(X)$. \\ 
     Show that  $P$ is regular if and
     only if $\rank((\frac{\partial f_i}{\partial
       x_j}(\underline{p}))_{i,j})=n-d$. \\
     {\footnotesize (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 $P$!)}
   \end{exercise}

   \begin{exercise}
     Let $C$ be a non singular projective plane curve over the algebraically
     closed field $K$. Show that $C$ is irreducible.\\
     {\footnotesize (Hint, use exercise 20 b.)}
   \end{exercise}

   \begin{exercise}
     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?\\
     {\footnotesize (Hint, you may use Singular to find the generating ideal
       of $X$ resp.\ to verify your own result.)}
   \end{exercise}

   \begin{exercise}
     Let $K$ be an algebraically closed field, $X$ a variety over $K$
     and $H$ a hypersurface in $X$ (i.\ e.\ a puredimensional subvariety of
     codimension 1). Let $x\in X$ such that $\ko_{X,x}$ is a unique
     factorization domain and that $x$ is a regular point of $H$.\\
     Show that $x$ is also a regular point of $X$. Is the assumption
     that $\ko_{X,x}$ be factorial necessary?\\
     {\footnotesize (Hint, reduce to the affine case and show that the Jacobi
       rank of the defining ideal $I_x$ for $H$ in $x$ is at most
       1. For this use as in exercise 9 the algebraic
       result: \textit{A noetherian integral domain is a unique
         factorization domain if and only if every prime ideal of
         height 1 is principal.})}
   \end{exercise}


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