## Hiroshima Mathematical Journal

### Rational curves on a smooth Hermitian surface

Norifumi Ojiro

#### 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
Revised: 1 February 2019
First available in Project Euclid: 6 April 2019

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

Digital Object Identifier
doi:10.32917/hmj/1554516042

Mathematical Reviews number (MathSciNet)
MR3936652

Zentralblatt MATH identifier
07090068

#### 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