Speaker: Alex Degtyarev.
Date: 10:30 21/08/2020
Title: Counting 2-planes in cubic 4-folds in P^5.
Abstract:
We use the global Torelli theorem for cubic 4-folds (C. Voisin) to establish the upper bound of
405 2-planes in a smooth cubic 4-fold. The only champion is the Fermat cubic.
We show also that the next two values taken by the number of 2-planes are 357 (the champion for
the number of *real* 2-planes) and 351, each realized by a single cubic. To establish the bound(s),
we embed the appropriately modified lattice of algebraic cycles to a Niemeier lattice and estimate the
number of square 4 vectors in the image. The existence is established my means of the surjectivity of
the period map. According to SchÃ¼tt and Hulek, the second best cubic with 357 planes can be realized
as a hyperplane section of the Fermat cubic in P^6.
(work in progress joint with I. Itenberg and J.Ch. Ottem)