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!

1$\begingroup$ What if you had, say, a rational elliptic surface $Y$ with a reducible fibre containing a rational curve over a point $p_0$, then pulled back via a highdegree covering $C \rightarrow P^1$ that was unramified over $p_0$? $\endgroup$– BertieMar 3, 2017 at 17:00

$\begingroup$ @Bertie  OOPS! I've forgotten to mention that the surface is minimal. $\endgroup$– Yuri ZarhinMar 3, 2017 at 17:25

$\begingroup$ Dear Zarhin: yes, I figured you meant minimal. But in fact I was talking about a reducible fibre which is a bunch of $2$ curves.Anyway, abx seems to have settled the matter to your satisfaction. $\endgroup$– BertieMar 4, 2017 at 22:34

1$\begingroup$ Dear Bertie, thank you for your example, $\endgroup$– Yuri ZarhinMar 11, 2017 at 3:59
1 Answer
I don't think this is true. Take a general pencil of cubics in $\mathbb{P}^2$, and blowup 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.