Some non-special cubic fourfolds

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.