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The Fano variety of lines and rationality problem for a cubic hypersurface
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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.
Forward citations
Cited by 2 Pith papers
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Zeta functions of K3 categories over finite fields
Noncommutative K3 surfaces over finite fields get zeta functions whose point counts can be negative, obstruct geometricity, and in one explicit example, perfectly mimic a K3 surface without being geometric.
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Derived categories of Fano varieties of lines
Proves Galkin's derived-equivalence conjecture for generic cubic fourfolds in Hassett divisors with d/2 a perfect square, and establishes the associated weight-two Hodge isometry for all smooth cubic fourfolds.
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