Some Non-Special Cubic Fourfolds

Abstract

In [20], 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 [20] are not Noether–Lefschetz divisors.