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(Fq2). 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.
|Journal||Moscow Mathematical Journal|
|Publication status||Published - 2020|
- Hermitian surfaces
- Cubic surfaces
- Intersection of surfaces
- Rational points