Kodaira 维数为一的 Jacobian 椭圆曲面上的半丰性
Semiampleness on Jacobian elliptic surfaces of Kodaira dimension one
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中文总结 AI 辅助
本文研究 Kodaira 维数为一的 Jacobian 椭圆曲面，在可约纤维条件下给出 Mori dream 曲面的判据，并构造反例说明奇数纤维时结论不成立。
中文摘要 AI 辅助
设 $\pi: X\to \mathbb P^1$ 是 $\mathbb C$ 上的半稳定 Jacobian 椭圆曲面，并设 $\chi=\chi(\mathcal O_X)\ge3$，从而 $\kappa(X)=1$。假设 $\pi$ 的 Mordell-Weil 群有限，且 $\pi$ 至少有一个可约纤维，可约纤维的类型为 $I_{n_1},\cdots, I_{n_s}$。最近，Laface 等人证明了零截面和可约纤维的分量生成 $\overline{\mathrm NE}(X)$ 当且仅当 $$\delta(\pi):=\sum_{i=1}^s\frac{\lfloor n_i^2/4\rfloor}{n_i}\le\chi.$$ 特别地，在此范围内 Mori 锥是有理多面体锥。他们还证明了 $N(\pi)=\sum_{i=1}^s n_i\le 2\chi+3$ 蕴含 $X$ 是 Mori dream 曲面。本文研究在 $N(\pi)\ge 2\chi+4$ 条件下 Mori dream 曲面的存在性问题。假设 $\delta(\pi)\le \chi$。我们首先证明 $X$ 上的每个 nef 迷向除子都是半丰的，并且当每个 $n_i$ 为偶数时成立。进一步，当每个 $n_i$ 为偶数时，我们得到 $X$ 为 Mori dream 曲面的判据：若 (i) 所有 $n_i=2$，或 (ii) $N(\pi)\le 2\chi+4$，则 $X$ 是 Mori dream 曲面；若某个 $n_j=4$，则 $X$ 是 Mori dream 曲面当且仅当 $N(\pi)\le 2\chi+4$。然而，一旦某个 $n_i$ 为奇数，我们构造一个 Jacobian 椭圆曲面 $\pi: Y\to\mathbb P^1$，满足 $\delta(\pi)=\chi=3$，Mordell-Weil 群平凡，奇异纤维构型为 $I_4+3I_3+23I_1$，使得八个非垂直迷向极值射线中没有一个是半丰的。特别地，$Y$ 不是 Mori dream 曲面。
英文摘要
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.
发表机构
- Universidad de Concepción（康塞普西翁大学）
- East China University of Science and Technology（华东理工大学）
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