Doubt regarding cross sections of a Killing horizon

Question

I was reading Rácz and Wald's paper, 'Extensions of spacetimes with Killing Horizons' and they are frequently referring to a cross section of a Killing horizon (a three surface where the norm of a timelike-at-infinity Killing vector vanishes). I initially thought that by the cross section, they meant a two surface which intersects the null generators of the Killing horizon at a fixed parameter length. However, in section 3 while proving a proposition, they chose a 'local' cross section that is 'sufficiently small', through a point  p of the horizon. This means my intuition is probably incorrect because I really don't understand how can you construct a 'small' cross section of a three surface as it would necessarily have to intersect all the null generators. Can someone please explain what is the exact meaning of their statement? Here is a link to the paper if anyone needs it: https://iopscience.iop.org/article/10.1088/0264-9381/9/12/008