The equations of general Hassett maximal cubic fourfolds

TL;DR

This paper constructs an explicit 16-dimensional component of Hassett maximal cubic fourfolds, analyzing their lattice structures and algebraic properties to confirm their maximality.

Contribution

It provides an explicit construction and analysis of a maximal-dimensional locus of Hassett maximal cubic fourfolds using algebraic and arithmetic methods.

Findings

Constructed an explicit irreducible component of maximal dimension sixteen.

Proved the primitive image of the lattice spans the entire Hassett subset.

Confirmed Hassett maximality through lattice and form analysis.

Abstract

In this note, we discuss Hassett maximal cubic fourfolds and construct an explicit irreducible component of maximal dimension sixteen of the locus $\mathcal{Z}$ of Hassett maximal cubic fourfolds. We utilize algebraic and arithmetic methods to analyze the associated lattice of these fourfolds. % By studying general integral quadratic forms and proving the ADC property for a specific ternary form, we demonstrate that the primitive image of our lattice spans the entire Hassett subset, confirming the Hassett maximality of the cubic fourfolds we describe.