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   \Blatt{14}
   \Abgabe{18/02/2002}

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


   \Nr
   Let $\rho_d:\P_K^n\rightarrow \P_K^N$ be the Veronese embedding and
   let $X=\im(\rho_d)\subset\P_K^N$. Show that $\deg(X)=d^n$.
   
   {\scriptsize Hint, show that $K[X]\cong\bigoplus_{k\geq 0} M_k$ as
     graded $K[y_0,\ldots,y_N]$-modules, where
     $M_k=K[x_0,\ldots,x_n]_{dk}$. Thus you may compute
     $P_{K[X]}(k)=\dim_K(M_k)$ explicitely. }


   \Nr
   Let $\sigma:\P_K^r\times\P_K^s\rightarrow \P_K^N$ be the Segre embedding and
   let $X=\im(\sigma)\subset\P_K^N$. Show that $\deg(X)=\binom{r+s}{s}$.

   {\scriptsize Hint, consider $\bigoplus_{k\geq 0}M_k$ with
     $M_k=\langle x_{i_1}y_{j_1}\cdots x_{i_k}y_{j_k}\; |\; 0\leq
     i_\mu\leq r, 0\leq j_\nu\leq s\rangle_K$, i.~e.~$M_k$ is the
     $K$-vector space of bihomogeneous polynomials of bidegree $(k,k)$.}

   \Nr
   Let $X\subset\P_K^n$ be a complete intersection of dimension $n-r$,
   i.~e.~$I(X)=\langle f_1,\ldots,f_r\rangle$, and let $d_i=\deg(f_i)$. Show that
   $\deg(X)=d_1\cdots d_r$.

   {\scriptsize Hint, let $X'=V(f_1,\ldots,f_{r-1})$ and consider the
     exact sequence $\xymatrix@C0.6cm{0\ar[r]& K[X']\ar[r]^{\cdot f_r} &
       K[X']\ar[r] &K[X]\ar[r] &0}$, using induction on $r$.}
   

   \Nr
   Let $f,g\in K[x,y]\setminus K$ with\footnote{For $f\in K[x,y]$ we
     have $\mult_p(f)=\max\{k\geq 0\;|\; f\in \m_p^k\}$.}
   $\mult_p(f)=m$ and
   $\mult_p(g)=n$. Show that $i\big(V(f),V(g);p\big)\geq m\cdot n$.

   {\scriptsize Hint, consider the exact sequence
     $\xymatrix@C0.8cm{\ko_{\A^2_K,p}/\m_p^n\oplus
       \ko_{\A^2_K,p}/\m_p^m \ar[r]^{\hspace*{0.6cm}(f,g)} &
       \ko_{\A^2_K,p}/\m_p^{m+n}\ar[r] &
       \ko_{\A^2_K,p} /\big(\m_p^{m+n}+\langle f,g\rangle\big)
       \ar[r] & 0}$
     and note that $i\big(V(f),V(g);p\big)$ is an upper bound for the
     dimension of the last $K$-vector space.}
 

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