Pairwise incident planes and hyperkaehler four-folds
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.
Cite
@article{arxiv.1204.6257,
title = {Pairwise incident planes and hyperkaehler four-folds},
author = {Kieran G. O'Grady},
journal= {arXiv preprint arXiv:1204.6257},
year = {2012}
}