Plane sections of Fermat surfaces over finite fields

In this paper, we characterize all curves over Fq arising from a plane sectionP:X3−e0X0−e1X1−e2X2=0 of the Fermat surfaceS:X0d+X1d+X2d+X3d=0, where q=ph=2d+1 is a prime power, p>3, and e0,e1,e2∈Fq. In particular, we prove that any nonlinear component G⊆P∩S is a smooth classical curve of degree n⩽d attaining the Stöhr-Voloch bound#G(Fq)⩽12n(n+q−1)−12i(n−2), with i∈{0,1,2,3,n,3n}.