I need a reference for the following assertion (if it's true). Let $X$ be a minimal elliptic surface over the field of complex numbers. Assume that its Kodaira dimension $\kappa(X)=1$. Then $X$ does not contain rational curves. Thanks!



Rational curves on elliptic surfaces
I don't think this is true. 
Take a general pencil of cubics in $\mathbb{P}^2$, and blow-up the 9 fixed points to get an elliptic fibration $f:S\rightarrow \mathbb{P}^1$ which admits a section (at least 9 in fact). Pull back by a degree $n\geq 3$ covering $\mathbb{P}^1\rightarrow \mathbb{P}^1$ branched along two points $p\_1,p\_2$ such that $\ f^{-1}(p\_i)\ $ is smooth. You get a new fibration $f':S'\rightarrow \mathbb{P}^1$ with $\kappa (S')=1$, $S'$ minimal; the sections of $f$ pull back to sections of $f'$ which are smooth rational curves.


