Past exam of the short options course of the University of Oxford / 2021 / 3 / a

A projective transformation of $\mathbb{P}({\mathbb{F}}^{n+1})$ is induced by an invertible linear map $A:{\mathbb{F}}^{n+1}\to {\mathbb{F}}^{n+1}$:This is well defined because replacing $v$ by a nonzero scalar multiple does not change $[Av]$. Multiplying $A$ by a nonzero scalar also leaves $\tau$ unchanged. Conversely these are precisely the transformations in the projective linear group ${\mathrm{PGL}}_{n+1}(\mathbb{F})$. No field automorphism is included in this definition.

Choose nonzero vectors $p_i\in {\mathbb{F}}^2$ representing $P_i$, and write $[u,v]=\det (u,v)$. Use the cross-ratio conventionDistinctness makes the denominator nonzero. Every rescaling of a representative occurs once in numerator and denominator, so the value depends only on the projective points. Swapping both pairs simply exchanges the numerator's factors and the denominator's factors:Under an invertible linear map $A$, $[Au,Av]=\det (A)[u,v]$. Two factors of $\det (A)$ occur above and below the fraction, proving projective transformation invariance.

Finally choose homogeneous coordinates making $P_0=[1:0]$, $P_1=[0:1]$, $P_2=[1:1]$. This normalization exists by the allowed theorem about a projective frame. A point distinct from $P_0$ is uniquely $[t:1]$, and the displayed cross-ratio is $t$. Thus equality of the two cross-ratios forces equal coordinates andThis also shows that for fixed first three distinct points the fourth point is determined by its cross-ratio.