Speaker: Sergey Galkin

Title: Cubics: lines, squares, and irrationality

Date: March 24, 2014, 15:30 - 16:30

Place: locally free geometry seminar, King's College London

I will describe our joint work with Evgeny Shinder. We prove that generic cubic fourfold is irrational under the assumption that the class of an affine line is not a zero divisor in the Grothendieck ring of complex varieties. Main new geometric ingredient of the proof is a beautiful formula, that relates classes of a cubic hypersurface itself, its symmetric square, variety of lines and the singular locus. The formula is unconditional and holds over any reduced cubic hypersurface over arbitrary field. It also gives another proof for the theorem of Cayley about 27 lines on a surface, as well as many similar results for singular surfaces over non-algebraically closed fields.