Lefschetz pencils on a certain hypersurface in positive characteristic

Abstract

We examine Lefschetz pencils of a certain hypersurface in ${\bf P}^{3}$ 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 ${\bf F}_{q}$ (resp. over ${\bf F}_{q^{2}}$) and the lines over ${\bf F}_{q}$ (resp. over ${\bf F}_{q^{2}}$) with $q = p^{a}$.