Weighted projective space

Question

Here is one example from Cox's lecture notes:

I really don't know how to use $M$ to define an automorphism. Intuitively, I can rescale the first coordinate to $1$ and I think the isomorphism is true. But I don't understand his method.

Answer 1

I guess that it should be something like this: take an element

$$ M=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in GL_2(\mathbb{Z}) $$ then we use it to define an automorphism of $(\mathbb{C}^*)^2$ by putting

$$ \begin{pmatrix} t_1 \\ t_2 \end{pmatrix} \mapsto \begin{pmatrix} t_1^a t_2^{-b} \\ t_1^ct_2^{-d} \end{pmatrix} $$

(I don't know how to make proper parentheses for the matrices, sorry)