Pairwise incident planes and hyperkaehler four-folds

TL;DR

This paper investigates the maximum size of finite families of pairwise incident planes in complex projective spaces, linking them to special sextic hypersurfaces and hyperkaehler four-folds, and establishes an upper bound of 20 with a conjecture of 16.

Contribution

It introduces new bounds on the size of such families and connects geometric configurations to hyperkaehler four-folds and EPW sextic hypersurfaces.

Findings

Maximum size of such families in 5-space is at most 20.

Existence of families with 16 planes is demonstrated.

Families spanning 6-dimensional space are classified as elementary.

Abstract

We address the following question: what are the cardinalities of maximal finite families of pairwise incident planes in a complex projective space? One proves easily that the span of the planes has dimension 5 or 6. Up to projectivities there is one such family spanning a 6-dimensional projective space - this is an elementary result. Maximal finite families of pairwise incident planes in a 5-dimensional projective space are considerably more misterious: they are linked to certain special (EPW) sextic hypersurfaces which have a non-trivial double cover, generically a hyperkaehler 4-fold. We prove that the cardinality of such a set cannot exceed 20. We also show that there exist such families of cardinality 16 - in fact we conjecture that 16 is the maximum.