# Some non-special cubic fourfolds

## TL;DR

This paper provides an independent proof that certain divisors in the moduli space of cubic fourfolds are not Noether-Lefschetz divisors, using explicit examples and point counting over finite fields.

## Contribution

It offers a new proof and explicit examples demonstrating that some divisors previously thought to be special are actually not Noether-Lefschetz divisors.

## Key findings

- Explicit cubic fourfold in the divisor is Noether-Lefschetz general
- Two other divisors are shown not to be Noether-Lefschetz divisors
- Point counting methods over finite fields are used for proof

## Abstract

In [1309.1899], Ranestad and Voisin showed, quite surprisingly, that the divisor in the moduli space of cubic fourfolds consisting of cubics "apolar to a Veronese surface" is not a Noether-Lefschetz divisor. We give an independent proof of this by exhibiting an explicit cubic fourfold X in the divisor and using point counting methods over finite fields to show X is Noether-Lefschetz general. We also show that two other divisors considered in [ibid.] are not Noether-Lefschetz divisors.