TITLE: Is there any rational curve on an Abelian variety?
QUESTION [12 upvotes]: Is that true that there is no rational curves contained in an Abelian variety? If it's true,  is that because abelian varieties are not uniruled? How do I know whether an abelian variety is not uniruled?

REPLY [2 votes]: Assume there is a rational curve $C$ in an Abelian variety $A$. Let $p\in C$ be a point and $q\in A$ another point. There exists an automorphism $\sigma:A\rightarrow A$ such that $\sigma(p) = q$. Then $\Gamma = \sigma(C)$ is a rational curve through $q$. In this way we see that $A$ is covered by rational curves and its Kodaira dimension is negative. A contradiction because $k(A) = 0$. We conclude that there are not rational curves on an Abelian variety.