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.

arXiv commentSome typos fixed, references added, exposition improved

References in corpus (8)

• Derived categories of cubic fourfolds
• Hodge theory and derived categories of cubic fourfolds
• Derived categories of cubic and V14 threefolds
• Real Algebraic Surfaces
• On the Jordan-Hölder property for geometric derived categories
• A simple counterexample to the Jordan-Hölder property for derived categories
• Birational self-maps and piecewise algebraic geometry
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Cited by in corpus (19)

• The K3 category of a cubic fourfold
• Cremona transformations and derived equivalences of K3 surfaces
• The generalized Franchetta conjecture for some hyper-Kähler varieties, II
• The class of the affine line is a zero divisor in the Grothendieck ring: via $G_2$-Grassmannians
• Cycles, derived categories, and rationality
• Derived categories of flips and cubic hypersurfaces
• Big quantum cohomology of Fano complete intersections
• L-equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces
• Automorphisms of positive entropy on some hyperKahler manifolds via derived automorphisms of K3 surfaces
• Twisted cubics and quadruples of points on cubic surfaces
• Lagrangian embeddings of cubic fourfolds containing a plane
• Motives of moduli spaces on K3 surfaces and of special cubic fourfolds
• Factorization centers in dimension two and the Grothendieck ring of varieties
• Categorical dimension of birational automorphisms and filtrations of Cremona groups
• The standard conjectures for the variety of lines on a cubic hypersurface
• Some nilpotence theorems for algebraic cycles
• Hilbert schemes of two points on K3 surfaces and certain rational cubic fourfolds
• Motivic limits for Fano varieties of $k$-planes
• Nonrationality of a generic cubic fourfold