Lefschetz pencils on a certain hypersurface in positive characteristic

We examine Lefschetz pencils of a certain hypersurface in P-3 over an algebraically closed field of characteristic p > 2, and determine the group structure of sections of the fiber spaces derived from the pencils. Using the structure of a Lefschetz pencil, we give a geometric proof of the unirationality of Fermat surfaces of degree p(a) + 1 with a positive integer a which was first poved by Shioda [10]. As byproducts, we also see that on the hypersurface there exists a (q(3) + q(2) + q + 1)(q+1)-symmetric configuration (resp. a ((q(3) + 1)(q(2) + 1)(q+1), (q(3) + 1)(q + 1)q(2)+1)-configuration) made up of the rational points over F-q (resp. over Fq(2)) and the lines over F-q (resp. over Fq(2)) with q = p(a).