contradicting the assumption that $\pi_1(X)^{\rm alg} = (1)$.

On the other hand, by the canonical divisor formula, we have $K_X = \pi^*(K_{{\bf P}^1}) + \chi({{{\mathcal{O}}}_X}) F_1
+ \sum_{i=1}^t (m_i - 1) (F_i)_{\rm red}
\sim_{\bf Q} (-1 + \sum_{i=1}^t (1 - \frac{1}{m_i})) F_1$ (so $\pi$ is the only elliptic fibration on $X$). Since $\kappa(X) = 1$, we see that $t \ge 2$. Now the lemma follows from the results above.

\(2) Since $\Gamma$ is of elliptic fibre type, $0 = K_X . \Gamma = \Gamma^2 = 0$. Hence $\Gamma$ is a multiple of a fibre of $\pi$. Since the support of $\Gamma$ ($< D$) is a tree, it is of type $I_n^*$, $II^*$, $III^*$ or $IV^*$, whence $\Gamma$ is a full fibre (and is not a multiple fibre). By the assumption, there is an $E$ in $D$ such that ${\text{\rm Supp}}(E + \Gamma)$ is a connected rational tree. Thus $E . \Gamma \le 6$ and the equality holds if and only if $\Gamma$ is of type $II^*$ and $E$ meets the coefficient-6 component of $\Gamma$. Now (2) follows from (1).

The example below shows that an assumption like the condition (3) in Conjecture \[mainconj\] might be necessary.

\[example\] We shall construct 