## Hiroshima Mathematical Journal

- Hiroshima Math. J.
- Volume 46, Number 2 (2016), 195-215.

### 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.

#### Article information

**Source**

Hiroshima Math. J., Volume 46, Number 2 (2016), 195-215.

**Dates**

Received: 25 August 2015

Revised: 29 February 2016

First available in Project Euclid: 12 August 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.hmj/1471024949

**Digital Object Identifier**

doi:10.32917/hmj/1471024949

**Mathematical Reviews number (MathSciNet)**

MR3536996

**Zentralblatt MATH identifier**

1354.14067

**Subjects**

Primary: 14J70: Hypersurfaces

Secondary: 14J50: Automorphisms of surfaces and higher-dimensional varieties

**Keywords**

Degeneration Fermat hypersurface positive characteristic

#### Citation

Hoai Hoang, Thanh. Degeneration of Fermat hypersurfaces in positive characteristic. Hiroshima Math. J. 46 (2016), no. 2, 195--215. doi:10.32917/hmj/1471024949. https://projecteuclid.org/euclid.hmj/1471024949