- #1

gurilupi

- 7

- 0

$$\begin{pmatrix}

X_1 & X_2 & X_3\\

X_2 & X_3 & X_4

\end{pmatrix}.$$

I have to find a Noether normalization ##k[Y_1, Y_2, Y_3, Y_4] \subseteq k[X_1, X_2, X_3, X_4]## with ##I \cap k[Y_1, Y_2, Y_3, Y_4] = (Y_1, \ldots, Y_r)## for a suitable ##r##.

I've done: The maximal minors are the determinant(s) of the largest submatrices, i.e. in this case all ##2 \times 2## submatrices which then are (by deleting a column): ##\begin{pmatrix}

X_1 & X_2\\

X_2 & X_3

\end{pmatrix}, \begin{pmatrix}

X_1 & X_3\\

X_2 & X_4

\end{pmatrix}, \begin{pmatrix}

X_2 & X_3\\

X_3 & X_4

\end{pmatrix}.## Then, taking determinants we get ##I = (X_1 X_3 - X_{2}^{2}, X_1 X_4 - X_3 X_2, X_2 X_4 - X_{3}^{2})##.

The next step would be to use the constructive proof of Noether's Normalization Lemma. However, I can't seem to understand the entire procedure of that proof and how to apply it to this problem. Perhaps if someone can illustrate this process, then I will better understand it after seeing it done.