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arXiv · 1405.5154

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

Sergey Galkin·Evgeny Shinder
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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.

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Sergey Galkin, Evgeny Shinder. 2014-06-25. The Fano variety of lines and rationality problem for a cubic hypersurface. https://arxiv.org/abs/1405.5154

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