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Nicolas Addington : Cubic fourfolds and K3 surfaces

Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics -- conjecturally, the ones which are rational -- have K3s associated to them geometrically. Hassett has studied the cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied the cubics with K3s associated to them at the level of derived categories. These two notions of having an associated K3 should coincide. We prove they coincide generically. That is, Hassett's cubics form a countable union of irreducible Noether-Lefschetz divisors in moduli space, and Kuznetsov's cubics are a dense subset of these, forming a non-empty, Zariski open subset in each divisor.

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