FPT algorithm for point line cover
The algorithm should say "choose $p \in S'$", that is, the text is the correct version, rather than the pseudocode.

Note how the algorithm works:


$S',S_{1},\ldots,S_{k}$ form a partition of $S$ (i.e. every element of $S$ is in exactly one of $S',S_{1},\ldots,S_{k}$).
At each step it chooses which $S_{i}$ to put $p$ into.
Initially $S' = S$, and $S_{i} = \emptyset$ for all $i$.


So $S'$ is the set of points that still need to be covered, and $S_{i}$ is the set of points covering by hyperplane $i$, in some abstract sense. Thus the algorithm, at each iteration, picks an uncovered element from $S'$, and puts it in some $S_{i}$.

Interestingly, this error made it all the way to the journal version.