Projective plane curves over a finite field with conical components

Abstract

We present the list of maximal projective plane curves containing conics and those which are arrangements of conics. The number of rational points and the corresponding polynomials are given. The third highest number of points of projective curves of degree d over a finite field Fq (d < [q/3]) is associated only to some linear curves. We show that for q/2 + 2 < d < q, this is no longer the case: the third highest number of points can also be obtained by some curves containing a conic. Throughout this work, we obtain some bounds concerning the number of Fq-points of curves with linear, conic and cubic factors. these bounds apply (not sharply) to irreducible curves