On lattice polarizable cubic fourfolds

TL;DR

This paper extends the understanding of Hassett divisors in the moduli space of cubic fourfolds, providing new tools for analyzing their intersections, and introduces criteria for algebraic cohomology related to associated K3 surfaces.

Contribution

It generalizes properties of Hassett divisors to higher rank lattices, offers an algorithm for their intersection components, and constructs examples of rational cubic fourfolds.

Findings

Fermat cubic fourfold lies in every Hassett divisor.

Developed an algorithm to determine irreducible components of divisor intersections.

Provided a numerical criterion for algebraic cohomology related to K3 surfaces.

Abstract

We extend non-emtpyness and irreducibility of Hassett divisors to the moduli spaces of $M$-polarizable cubic fourfolds for higher rank lattices $M$, which in turn provides a systematic approach for describing the irreducible components of intersection of Hassett divisors. We show that Fermat cubic fourfold is contained in every Hassett divisor, which yields a new proof of Hassett's existence theorem of special cubic fourfolds. We obtain an algorithm to determine the irreducible components of the intersection of any two Hassett divisors and we give new examples of rational cubic fourfolds. Moreover, we derive a numerical criterion for the algebraic cohomology of a cubic fourfold having an associated K3 surface and answer a question of Laza by realizing infinitely many rank $11$ lattices as the algebraic cohomologies of cubic fourfolds having no associated K3 surfaces.