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

2014/05/20 by Galkin, Sergey, Shinder, Evgeny · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1405.5154

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.

Cited by

• L-equivalence and Fourier--Mukai partners of cubic fourfolds
• Some remarks on L-equivalence for cubic fourfolds and hyper-Kähler manifolds

Related

• Hilbert schemes of two points on K3 surfaces and certain rational cubic fourfolds
• The standard conjectures for the variety of lines on a cubic hypersurface
• Lines on cubic threefolds and fourfolds containing a plane
• The degree of irrationality of hypersurfaces in various Fano varieties
• Derived categories of Fano varieties of lines