Intersection of irreducible curves and the Hermitian curve

Let Hq denote the Hermitian curve in P² over Fₐℂ and Cd be an irreducible plane projective curve in P² also defined over Fₐℂ of degree d. Can Hq and Cd intersect in exactly d (q+1) distinct Fₐℂ-rational points? B\'ezout's theorem immediately implies that Hq and Cd intersect in at most d (q+1) points, but equality is not guaranteed over Fₐℂ. In this paper we prove that for many d q²-q+1, the answer to this question is affirmative. The case d=1 is trivial: it is well known that any secant line of Hq defined over Fₐℂ intersects Hq in q+1 rational points. Moreover, all possible intersections of conics and Hq were classified by Donati et al. in 2009 and their results imply that the answer to the question above is affirmative for d=2 and q 4, as well. However, an exhaustive computer search quickly reveals that for (q, d) \ (2, 2), (3, 2), (2, 3) \, the answer is instead negative. We show that for q d q²-q+1, d= (q+1) /2 and d=3, q 3 the answer is again affirmative. Various partial results for the case d small compared to q are also provided.