A smooth geometrically connected curve over the finite field [Formula: see text] with gonality has at most rational points. Faber and Grantham conjectured that there exist curves of every sufficiently large genus achieve this bound. In paper, we show bound can be achieved for an infinite sequence genera using abelian covers projective line. We also argue will not suffice to prove full conjecture.

Let R and B be sets of n red and n blue points in the plane, respectively, with P = R∪ B. Let M be a perfect matching between points from R and B, using n straight line segments to match the points, that is, each point is an endpoint of exactly one line segment, and each line segment has one red and one blue endpoint. We forbid line segments to cross. Denote the length of a longest line segment...