Speaker: Sergey Galkin
Title: Lines on rational cubic fourfolds, and associated K3 surfaces
Date: May 21, 2014, 15:30 - 17:00
Place: Laboratory of Algebraic Geometry, HSE, Moscow
This is a joint work with Evgeny Shinder. It is expected (after Iskovskikh, Zarkhin, Tregub, Beauville-Donagi, Hassett, Kulikov, Kuznetsov, Addington-Thomas, and others) that generic cubic fourfolds are irrational, and rational ones are related in some way to K3 surfaces. For example, Pfaffian cubics were shown to be rational by Morin in 1940, and in 1984 Beauville and Donagi shown that their variety of lines is a Hilbert scheme of 2 points on a K3 surface, related to the original cubic by projective duality. We generalize this result to all rational cubic fourfolds, under the assumption of Denef-Loeser's conjecture. Namely, if a class of an affine line is not a zero divisor in the Grothendieck ring of varieties, then Fano variety of lines on a rational cubic fourfold is birational to a Hilbert scheme of two points on some K3 surface. The proof uses a theorem of Larsen and Lunts and a new unconditional relation between the classes of the variety of lines and of the symmetric square of any cubic hypersurface in the Grothendieck ring of varieties. The latter relation also reproduces many known results, such as 27 lines on a cubic surface.