# Projectivity maps that fix three points

Please check this definition: A projectivity is a bijection $PV\to PW$ induced by an isomophism $\phi: V\to W$ given by $\phi(kv)=k\phi (v)$. Now, i have seen here

Old Question

that an answer says a projectivity that fixes any three points is the identity. Can any one prove this rigorously? Does this hold in any dimension?

## 1 Answer

The statement only holds for the one-dimensional case, the projective line.

One way to prove it is by showing that the cross ratio remains invariant under projectivities, and verifying that given three points, the cross ratio already uniquely defines a fourth.

Another approach would be by observing that a projectivity in $d$ dimensions is uniquely defined by $d+2$ points and their images, as a generalization of this post. So if $d+2$ points are fixed, the uniquely defined projectivity has to be the identity. In the general case you have to require the preimage and image points to be in generic position, e.g. no three collinear. For $d=1$ this reduces to a requirement for the points to be distinct.