1$ and $c_2 \ge 2 + 0 d$, a contradiction.

Surfaces with negative $\kappa$, Part II: relatively minimal elliptic
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In this section we consider relatively minimal elliptic surface $\pi : X \rightarrow B$ with Kodaira dimension $\kappa(X)
= -\infty$. As far as the Conjecture \[mainconj\] is concerned, we may assume that the irregularity $q(X) = 0$ by virtue of Theorem \[irreg\]. So $X$ is a rational surface and $B = {\bf
P}^1$. By the canonical divisor formula, we see that $\pi$ has at most one fibre $F_0$ with multiplicity $m \ge 2$; moreover, such $F_0$ (if exists) is of Kodaira type $I_n$ ($n \ge 0$), and $-K_X
= (F_0)_{\rm red}$.

We show that Conjecture 1.1 is true if $K_X + L$ is nef and big (but without the assumption of the effectiveness of $L$):

\[ratell\] Let $\pi : X \rightarrow B$ be a relatively minimal elliptic surface with $\kappa(X) = -\infty$. Suppose that $L$ is a nef and big ${\bf Q}$-divisor such that $K_X
+ L$ is nef and big. Then $H^0(X, K_X + \lceil L \rceil) \ne 0$.

By Theorem \[irreg\], we may assume that $q(X) = 0$, so $B = {\bf P}^1$ and $X$ is a rational 