On lattice polarizable cubic fourfolds

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.

Cubic Fourfolds with an Involution

“…In other words, M M is the moduli space of cubics with primitive algebraic lattice equivalent to the prescribed lattice M . We say such a cubic is M -polarized [YY21]. Up to passing to a normalization, M M is (possibly the complement of some divisors in) a locally symmetric variety D M /Γ M , where D M is the type IV domain associated with the transcendental lattice T = M ⊥ L .…”

“…Further, the authors show that dim Z ≥ 13, by illustrating a lattice M of rank 7 such that the moduli of M -polarized cubic fourfolds M M is non-empty, and M contains a labeling of determinant d for every d > 6, d ≡ 0, 2 mod 6. We adapt the method in [YY21] to our situation.…”

“…For a cubic X with the involution φ 3 , we describe the lattice A(X) prim ∼ = M explicitly. We can consider the moduli space of cubic fourfolds with the involution φ 3 as M -polarized cubic fourfolds [YY21]. Denote this moduli space M φ3 ; it is irreducible and 10-dimensional.…”

“…The intersection of all Hassett divisors C d first appears in [YY21]. They exhibit a component of dimension at least 13 by lattice theoretic considerations.…”

“…They exhibit a component of dimension at least 13 by lattice theoretic considerations. However, the only geometric example of a Hassett maximal cubic, identified in [YY21], is the Fermat cubic. Our example of cubic fourfolds with anti-symplectic involution of type φ 3 gives a 10 dimensional geometrically meaningful locus of Hassett maximal cubics.…”