On two upper bounds for hypersurfaces involving a Thas' invariant

Let Xn be a hypersurface in Pn+1 with n≥1 defined over a finite field Fq of q elements. In this note, we classify, up to projective equivalence, hypersurfaces Xn as above which reach two elementary upper bounds for the number of Fq-points on Xn which involve a Thas' invariant.