Maximum number of points on intersection of a cubic surface and a non-degenerate Hermitian surface

In 1991 Sørensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree d and a non-degenerate Hermitian surface in P³(F_q2). The conjecture was proven to be true by Edoukou in the case when d = 2. In this paper, we prove that the conjecture is true for d = 3. For q ≥ 4, we also determine the second highest number of rational points on the intersection of a cubic surface and a non-degenerate Hermitian surface. Finally, we classify all the cubic surfaces that admit the highest and, for _q ≥ 4, the second highest number of points in common with a non-degenerate Hermitian surface. This classification disproves a conjecture proposed by Edoukou, Ling and Xing.