A note on plane pointless curves

Let d(q) denote the minimal degree of a smooth projective plane curve that is defined over the finite field Fq and does not contain Fq rational points. We are interested in the asymptotic behavior of d(q) for q→∞. To the best of the author's knowledge the problem of estimating the asymptotic behavior of d(q) was not considered previously. In this note we establish the following bounds:(1)14⩽lim̲q→∞logqd(q)⩽13. More specifically, for every characteristic p>3 we construct a sequence of pointless Fermat curvesxdk+ydk+zdk=0,over Fpmk, such that limk→∞logpmkdk=1/3.