The Fano variety of lines and rationality problem for a cubic hypersurface
By Sergey Galkin, Evgeny Shinder. 2014
61 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/005049d46635f6fb06b86e9ce30fabc3322519eb
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In Computational Theory and Mathematics
In Geometry and Topology
Citations in the corpus
- On two rationality conjectures for cubic fourfolds (2014, background): “. It is widely expected that a smooth complex cubic fourfold X is rational if and only if it has an associated K3 surface in the sense of Hassett [8] or Kuznetsov [11]. New work of Galkin and Shinder [7] suggests instead that if X is rational then the variety F of lines on X is birational to the Hil”
- The generalized Franchetta conjecture for some hyper-Kähler varieties, II (2020, methodology): “(i) The isomorphism (1) lifts the isomorphism in cohomology due to Galkin– Shinder [GS14, Th. 6.1] to the level of Chow motives.”