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 $(n+1)\times(n+1)$ matrix with coefficients $a_{ij}$ in $k$, and let $X_A$ be a hypersurface of degree $q + 1$ in the projective space $\mathbf{P}^n$ defined by $\sum a_{ij}x_i x^q_j=0$. 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.