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I am a little bit confused conerning the following example of projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$.

On the affine part $\mathbb K\subseteq \mathbb KP^1$ they are exactly the rational maps of the form $x\rightarrow \frac{ax+b}{cx+d}, x\in\mathbb K$

My question is: How can I wirte down explicitly the six projectivities of the form above that permutes the points $0,1,\infty$ ?

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up vote 2 down vote accepted

I dislike the formulation as a fraction, as it doesn't handle $\infty$ too well. I prefer homogenous coordinates and a matrix notation. Doesn't change the core of the issue, though.

\begin{align*} 0&\mapsto0 & 1&\mapsto1 & \infty&\mapsto\infty & \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} && x&\mapsto\frac{x}{1} \\ 0&\mapsto1 & 1&\mapsto0 & \infty&\mapsto\infty & \begin{pmatrix}-1 & 1 \\ 0 & 1\end{pmatrix} && x&\mapsto\frac{-x+1}{1} \\ 0&\mapsto\infty & 1&\mapsto1 & \infty&\mapsto0 & \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} && x&\mapsto\frac{1}{x} \\ 0&\mapsto\infty & 1&\mapsto0 & \infty&\mapsto1 & \begin{pmatrix}1 & -1 \\ 1 & 0\end{pmatrix} && x&\mapsto\frac{x-1}{x} \\ 0&\mapsto0 & 1&\mapsto\infty & \infty&\mapsto1 & \begin{pmatrix}1 & 0 \\ 1 & -1\end{pmatrix} && x&\mapsto\frac{x}{x-1} \\ 0&\mapsto1 & 1&\mapsto\infty & \infty&\mapsto0 & \begin{pmatrix}0 & 1 \\ -1 & 1\end{pmatrix} && x&\mapsto\frac{1}{-x+1} \\ \end{align*}

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