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 $\mathbb{P}^3(\mathbb{F}_{q^2})$. 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 $q \ge 4$, 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 $q \ge 4$, 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.