I just learned about the cross-ratio and that it is a projective invariant. I would like to use it to look at the curve defined over some algebraically closed field $k$ of characteristic $p>0$ given by $f(x,y)=y^p-y-x$, i.e. an Artin-Schreier extension of $k(x)$.
I don't understand how to compute the cross-ratio. This curve sits inside $\mathbb{P}^3_k$, so when I add/subtract points, do I do so component-wise? What does it even mean to multiply/divide points?
Any help would be very much appreciated!
Cross ratio is defined only for points in $P^1_k$. You can define it only for points in $P^n_k$ which belong to a common projective line. To define it on a projective line, use an affine patch containing all the points. Then check independence of the patch, using linear fractional transformations.