## Hiroshima Mathematical Journal

- Hiroshima Math. J.
- Volume 49, Number 1 (2019), 161-173.

### Rational curves on a smooth Hermitian surface

#### Abstract

We study the set $R$ of nonplanar rational curves of degree $d \lt q + 2$ on a smooth Hermitian surface $X$ of degree $q + 1$ defined over an algebraically closed field of characteristic $p > 0$, where $q$ is a power of $p$. We prove that $R$ is the empty set when $d \lt q + 1$. In the case where $d = q + 1$, we count the number of elements of $R$ by showing that the group of projective automorphisms of $X$ acts transitively on $R$ and by determining the stabilizer subgroup. In the special case where $X$ is the Fermat surface, we present an element of $R$ explicitly.

#### Article information

**Source**

Hiroshima Math. J., Volume 49, Number 1 (2019), 161-173.

**Dates**

Received: 18 July 2018

Revised: 1 February 2019

First available in Project Euclid: 6 April 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.hmj/1554516042

**Digital Object Identifier**

doi:10.32917/hmj/1554516042

**Mathematical Reviews number (MathSciNet)**

MR3936652

**Zentralblatt MATH identifier**

07090068

**Subjects**

Primary: 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx] 14M99: None of the above, but in this section

Secondary: 14N99: None of the above, but in this section

**Keywords**

rational curve Hermitian surface positive characteristic

#### Citation

Ojiro, Norifumi. Rational curves on a smooth Hermitian surface. Hiroshima Math. J. 49 (2019), no. 1, 161--173. doi:10.32917/hmj/1554516042. https://projecteuclid.org/euclid.hmj/1554516042