Combinatorics of disjoint linear subspaces of projective space, with an application to algebraic geometry

This talk is based on the joint seven author paper arXiv:2308.00761. We call a finite set of disjoint r dimensional planes in 2r+1 dimensional projective space (over any field K) an r-spread. I will explain how each r-spread has a canonical combinatorial structure which seems heretofore not to have been known and how this structure gives rise to a group for each spread. There are many open questions, for example, when is the group abelian? When is it finite? Very little work has been done for r > 1. For r=1, we have partial answers to these questions and in addition, we apply this structure to study the algebraic geometric notion of geproci subsets. (We say a finite subset Z of 3 dimensional projective space over an algebraically closed field is geproci if the projection of Z from a general point to a plane is a complete intersection. Nontrivial examples of geproci sets have been recognized only since 2019. The first example recognized was the set of 12 points coming from projectivizing the roots of the D_4 root system.)