Semiampleness on Jacobian elliptic surfaces of Kodaira dimension one

Source abstract

Let $π: X\to \mathbb P^1$ be a semistable Jacobian elliptic surface over $\mathbb C$, and set $χ=χ(\mathcal O_X)\ge3$, so that $κ(X)=1$. Assume that the Mordell-Weil group of $π$ is finite and that $π$ has at least one reducible fiber, the reducible fibers being of types $I_{n_1},\cdots, I_{n_s}$. Recently, Laface et al. proved that the zero section and the components of the reducible fibers generate $\overline{\mathrm NE}(X)$ if and only if $$δ(π):=\sum_{i=1}^s\frac{\lfloor n_i^2/4\rfloor}{n_i}\leχ.$$ In particular, the Mori cone is rational polyhedral in this range. They also proved that $N(π)=\sum_{i=1}^s n_i\le 2χ+3$ implies that $X$ is a Mori dream surface. In this paper, we study the existence problem of Mori dream surfaces provided that $N(π)\ge 2χ+4$. Suppose $δ(π)\le χ$. We first show that every nef isotropic divisor on $X$ is semiample, and whenever each $n_i$ is even. Furthermore, when every $n_i$ is even, we obtain criteria for $X$ to be a Mori dream surface: $X$ is a Mori dream surface provided that either (i) all $n_i=2$, or (ii) $N(π)\le 2χ+4$; if some $n_j=4$, then $X$ is a Mori dream surface if and only if $N(π)\le 2χ+4$. However, once some $n_i$ is odd, we construct a Jacobian elliptic surface $π: Y\to\mathbb P^1$ with $δ(π)=χ=3$, trivial Mordell-Weil group, and singular-fiber configuration $I_4+3I_3+23I_1$ for which none of the eight non-vertical isotropic extremal rays is semiample. In particular, $Y$ is not a Mori dream surface.