Cubic fourfolds behave in many ways like surfaces. Certain cubics—conjecturally, the ones that are rational—have specific surfaces associated to them geometrically. Hassett has studied cubics with surfaces associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with surfaces associated to them at the level of derived categories.
These two notions of having an associated 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.
"Hodge theory and derived categories of cubic fourfolds." Duke Math. J. 163 (10) 1885 - 1927, 15 July 2014. https://doi.org/10.1215/00127094-2738639