Some non-special cubic fourfolds

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.

13 pages, Macaulay2 and C++ code as ancillary files. Minor changes, to appear in Documenta Math

References in corpus (2)

• Transcendental obstructions to weak approximation on general K3 surfaces
• On the characteristic polynomial of the Frobenius on étale cohomology