The Fano variety of lines and rationality problem for a cubic hypersurface

Abstract

We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational to a Hilbert scheme of two points on a K3 surface; in particular, general cubic fourfold is irrational.

Some typos fixed, references added, exposition improved

References in corpus (2)

• Derived categories of cubic and V14 threefolds
• A simple counterexample to the Jordan-Hölder property for derived categories

Cited by in corpus (4)

• The class of the affine line is a zero divisor in the Grothendieck ring: via $G_2$-Grassmannians
• Automorphisms of positive entropy on some hyperKahler manifolds via derived automorphisms of K3 surfaces
• Lagrangian embeddings of cubic fourfolds containing a plane
• The standard conjectures for the variety of lines on a cubic hypersurface