Dichotomous point counts over finite fields

Abstract

We establish a near dichotomy between randomness and structure for the point counts of arbitrary projective cubic threefolds over finite fields. Certain "special" subvarieties, not unlike those in the Manin conjectures, play a distinguished role. This suggests that similar (near) dichotomies may exist more generally for other natural families of varieties.We also establish partial results for projective hypersurfaces in general, using tools including "worst-case" cohomological results of Skorobogatov or Katz, "average-case" results of Lindner, the existence theory of Lefschetz pencils, and the theory of discriminants. This continues a line of inquiry initiated by Hooley.