Invariantentheorie/8 3/Beispiel

${}D\longrightarrow K/n$

Operation von ${}\langle \xi _{n}\rangle$ auf ${}K^{n}$

1. in zwei Variablen ${}d(x)=1$ ${}d(y)=n-1$ ${}d(x)=???x$ ${}\xi$ operiert durch ${}{\begin{pmatrix}\xi &0\\0&\xi ^{n-1}\end{pmatrix}}$ Invariantenring:

   $v_{1}d_{1}+v_{2}d_{2}=0{\text{ in }}K/n$

   $v_{1}+v_{2}(n-1)=0{\text{ in }}K/n$

   Lösungen:

   ${\begin{array}{c}(n,0)\\(0,n)\\(1,1)\end{array}}\rightsquigarrow K[X^{n},Y^{n},XY]$

   $K[X^{n},Y^{n},XY]\cong K[U,V,W]/(UV-W^{n})$

   ${}A$-Singularitäten

   ${}A_{n-1}$-Quotientensingularitäten

2. ${}d(x)=1=d(y)$ in ${}K/n$ ${}{\begin{pmatrix}\xi &0\\0&\xi ^{n-1}\end{pmatrix}}$ ${}v_{1}+v_{2}=0$ in ${}K/n$ Lösungen:

   ${\begin{array}{c}(n,0)\\(n-1,1)\\\vdots \\(1,n-1)\\(0,n)\end{array}}\rightsquigarrow K[X^{n},X^{n-1}Y,X^{n-2}y^{2},\ldots ,y^{n}]$

   Veronesering ${}\rightsquigarrow$ nach Gleichungen