# Rational curves on a smooth Hermitian surface

## TL;DR

This paper investigates the existence and enumeration of nonplanar rational curves of certain degrees on smooth Hermitian surfaces over algebraically closed fields, revealing conditions for their existence and providing explicit examples.

## Contribution

It establishes the nonexistence of such curves for degrees less than q+1, counts the curves when degree equals q+1, and explicitly constructs an example on the Fermat surface.

## Key findings

- No rational curves of degree less than q+1 exist on the surface.
- The number of rational curves of degree q+1 is determined via automorphism group action.
- An explicit example of a rational curve on the Fermat surface is provided.

## Abstract

We study the set $R$ of nonplanar rational curves of degree $d<q+2$ 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 $d<q+1$. 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.