Hodge theory and derived categories of cubic fourfolds

TL;DR

This paper demonstrates that the Hodge-theoretic and derived category approaches to associating K3 surfaces with cubic fourfolds coincide generically, linking two different perspectives in algebraic geometry.

Contribution

It proves that Hassett's and Kuznetsov's notions of associated K3 surfaces for cubic fourfolds coincide on a dense, Zariski open subset of the moduli space.

Findings

Hassett's cubics form a countable union of Noether-Lefschetz divisors.

Kuznetsov's cubics are dense and form a non-empty, Zariski open subset within these divisors.

The two notions of association coincide generically in the moduli space.

Abstract

Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3s associated to them at the level of derived categories. These two notions of having an associated K3 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 non-empty, Zariski open subset in each divisor.