Cross ratio

I remember that in a complex analysis course and a combinatorics course, my teachers both discussed about cross ratio. I recently realized that Vakil's notes (19.9) contains a nice exposition about this, which I choose to regurgitate here with a bit more elementary explanations.

Let us work over a field $k.$ We can identify $$\mathrm{Aut}(\mathbb{P}^{n}) = \mathrm{PGL}_{n+1}(k) = \mathrm{GL}_{n+1}(k)/k^{\times}$$ because giving an automorphism $\pi : \mathbb{P}^{n} \rightarrow \mathbb{P}^{n}$ is the same as choosing a $k$-linear automorphism of $$H^{0}(\mathbb{P}^{n}, \mathscr{O}_{\mathbb{P}^{n}}(1)) = kx_{0} + \cdots + kx_{n}$$ up to nonzero global sections of $\mathbb{P}^{n},$ which consists of precisely $k^{\times}.$ If we consider this $k$-linear automorphism as $A \in \mathrm{PGL}_{n+1}(k),$ one can check that on $\mathbb{P}^{n}(k),$ the map $\pi$ is given by $x = [x_{0} : \cdots : x_{n}] \mapsto Ax.$ Here, we note that $x$ is seen as an $(n+1) \times 1$ column vector.

One can check that given any distinct $w, x, y \in \mathbb{P}^{1}(k),$ there is precisely one $A \in \mathrm{PGL}_{2}(k)$ such that $A : (w, x, y) \mapsto (0, 1, \infty),$ where we wrote $$a = [a : 1]$$ for any $a \in k = \mathbb{A}^{1}(k)$ and $\infty = [1:0].$ Even though it is a bit convoluted-looking, one can even write down explicitly what $A$ is. (Of course, it will be only unique up to a multiple of an element in $k^{\times}.$) The exact expression is the following:

$$A = \begin{bmatrix}x_{2}(w_{1}y_{2} - y_{1}w_{2}) & -x_{1}(w_{1}y_{2} - y_{1}w_{2}) \\ -w_{2}(y_{1}x_{2} - x_{1}y_{2}) & w_{1}(y_{1}x_{2} - x_{1}y_{2})\end{bmatrix},$$

where $w = [w_{1} : w_{2}]$ and similarly for $x$ and $y.$ Given any other $k$-point $z = [z_{1} : z_{2}]$ of the projective line, the cross ratio of the four points $w, x, y, z$ is defined as $Az.$

Motivation for the nomenclature. We have 5 cases.

Case 1. Consider the case where $w, x, y, z \neq \infty$ so that we can write $w = [w : 1]$ and similarly for $x, y, z.$ Then the cross ratio of $w, x, y, z$ is precisely

$$\frac{(w -y)(z - x)}{(w - z)(y - x)}.$$

Case 2. Let $w = \infty = [1 : 0].$ Then the cross ratio is

$$\frac{z - x}{y - x}.$$

Case 3. Let $x = \infty = [1 : 0].$ Then the cross ratio is

$$\frac{w - y}{w - z}.$$

Case 4. Let $y = \infty = [1 : 0].$ Then the cross ratio is

$$\frac{x - z}{w - z}.$$

Case 5. Let $z = \infty = [1 : 0].$ Then the cross ratio is

$$\frac{w - y}{x - y}.$$

Remark. Note that the cross ratios from Case 2, 3, 4, and 5 may us think that we took each letter to infinity, but we didn't! In any case, this is great because we can just remember the expression from Case 1 as it recovers every other case.

Writing $$\lambda = \lambda(w,x,y,z) = \frac{(w -y)(z - x)}{(w - z)(y - x)}$$ to mean the cross ratio of $w, x, y, z,$ note that

1. $\lambda(w,y,x,z) = 1 - \lambda,$
2. $\lambda(w,x,z,y) = 1/\lambda,$
3. $\lambda(w,z,x,y) = 1 - 1/\lambda,$
4. $\lambda(w,y,z,x) = 1/(1-\lambda),$
5. $\lambda(w,z,y,x) = \lambda/(1-\lambda).$