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

TL;DR

This paper explores the relationship between cubic hypersurfaces and their Fano varieties of lines, revealing implications for the rationality of cubic fourfolds through Grothendieck ring analysis.

Contribution

It establishes a link between the Fano variety of lines and the rationality problem for cubic fourfolds using Grothendieck ring techniques.

Findings

Fano variety of lines on a smooth rational cubic fourfold is birational to a Hilbert scheme of two points on a K3 surface.

General cubic fourfolds are shown to be irrational under certain conditions.

A relation between the class of a cubic hypersurface and its Fano variety in the Grothendieck ring is proven.

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.