Doubt regarding cross sections of a Killing horizon

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 the timelike-at-infinity Killing vector field vanishes). I initially thought that by a cross section, they meant a 2-surface which intersects the null generators of the Killing horizon at a fixed parameter length. However, in section 3 while proving a proposition, they made a choice of a local cross section through a point of the horizon that is 'sufficiently small'. This means my intuition is probably incorrect because I really don't understand how can you construct a 'small' cross section of a 3 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's a link to the paper if anyone needs it, https://iopscience.iop.org/article/10.1088/0264-9381/9/12/008.