$\Gamma$ is a full fibre of $\pi$ and of type $II^*$, $(m_1, m_2) = (2, 3)$ and $E . F = 6$ for some $E$ in $D$ (see [@BPV], Ch V, §7, for notation of singular fibres).

\(1) Since $\pi_1(X)^{\rm alg} = (1)$, we have $H_1(X, {\bold Z}) = (0)$ and hence $q(X) = 0$. So $\chi({{{\mathcal{O}}}_X}) = 1$. Since $\kappa(X) = 1$, there is an elliptic fibration $\pi : X \rightarrow \pi(X) = {\bf P}^1$, where the image is ${\bf P}^1$ because $q(X) = 0$. Let $F_i$ ($1 \le i \le t$) be all multiple fibres of $\pi$, with multiplicity $m_i$. If $m = gcd(m_1, m_2) \ge 2$, then the relation $m(F_1/m - F_2/m) \sim 0$ induces an unramified Galois ${\bf Z}/(m)$-cover of $X$, contradicting the assumption $\pi_1(X)^{\rm alg} = (1)$. If $t \ge 3$, then by Fox’s solution to Fenchel’s conjecture (see \[Fo\], \[Ch\]), there is a base change $B \rightarrow {\bold P}^1$ ramified exactly over $\pi(F_i)$ ($1 \le i \le t$) and with ramification index $m_i$. Then the normalization $Y$ of the fibre product $X \times_{{\bf P}^1} B$ is an unramified cover of $X$ (so that the induced fibration $Y \rightarrow B$ has no multiple fibres), again 