Dichotomous point counts over finite fields (Q6101535)

One says that a scheme \(X\) satisfies ``square root cancellation'' if the difference between the number of points over \(\mathbb F_{q^r}\) and the number of points of the corresponding projective space is of order \(q^{r\dim X/2}\). It is known that a ``square root cancellation'' may fail for a singular \(X\). The author, following the line of Hooley, investigates how frequently this is the case. The author shows that ``most'' hypersurfaces satisfy ``square root cancellation'' (Theorem 1.1). For a cubic hypersurfaces it is investigated when the ``square root cancellation'' fails (Theorems 1.2, 1.3). In the appendix some estimates about moments are obtained.