Degeneration of Fermat hypersurfaces in positive characteristic

Abstract

We work over an algebraically closed field k of positive characteristic p. Let q be a power of p. Let A be an (n + 1) × (n + 1) matrix with coefficients a ij in k, and let X A be a hypersurface of degree q + 1 in the projective space P n defined by a ij x i x q j = 0. It is well-known that if the rank of A is n + 1, the hypersurface X A is projectively isomorphic to the Fermat hypersuface of degree q + 1. We investigate the hypersurfaces X A when the rank of A is n, and determine their projective isomorphism classes.