Duke Mathematical Journal

Hodge theory and derived categories of cubic fourfolds

Nicolas Addington and Richard Thomas

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Abstract

Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics—conjecturally, the ones that are rational—have specific K3 surfaces associated to them geometrically. Hassett has studied cubics with K3 surfaces associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3 surfaces associated to them at the level of derived categories.

These two notions of having an associated K3 surface should coincide. We prove that they coincide generically: Hassett’s cubics form a countable union of irreducible Noether–Lefschetz divisors in moduli space, and we show that Kuznetsov’s cubics are a dense subset of these, forming a nonempty, Zariski-open subset in each divisor.

Article information

Source
Duke Math. J., Volume 163, Number 10 (2014), 1885-1927.

Dates
First available in Project Euclid: 8 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1404824304

Digital Object Identifier
doi:10.1215/00127094-2738639

Mathematical Reviews number (MathSciNet)
MR3229044

Zentralblatt MATH identifier
1309.14014

Subjects
Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
Secondary: 14J35: $4$-folds 14J10: Families, moduli, classification: algebraic theory 14J28: $K3$ surfaces and Enriques surfaces

Citation

Addington, Nicolas; Thomas, Richard. Hodge theory and derived categories of cubic fourfolds. Duke Math. J. 163 (2014), no. 10, 1885--1927. doi:10.1215/00127094-2738639. https://projecteuclid.org/euclid.dmj/1404824304


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