AI 中文总结
本文针对Kodaira维数为1的雅可比椭圆曲面，明确了其闭森锥有理多面体及森梦想曲面的纤维参数条件，构造了非森梦想的反例并对任意皮卡数ρ≥2构造了森梦想曲面。
AI 中文摘要
设π:X→ℙ¹是复数域ℙ¹上的雅可比椭圆曲面，满足χ=χ(𝒪_X)≥3，因此κ(X)=1。假设π的Mordell-Weil群有限，且π恰有一个可约纤维，类型为Iₙ。我们证明当n≤4χ时，X的闭森锥是有理多面体；当n≤2χ+3时，X是森梦想曲面。证明结合了森锥的显式描述与应用于其对偶丰饶射线零轨迹的Artin判别法。我们还证明，在Kodaira维数为1的情形下，Mordell-Weil群与自同构群均有限并不蕴含森锥的多面体性。在多面体范围内，我们构造了一个(χ,n)=(3,11)、皮卡数为12的雅可比椭圆曲面，其存在一个大且丰饶但非半ample的除子；特别地，该曲面不是森梦想曲面。最后，对每个整数ρ≥2，我们构造了一个Kodaira维数为1、皮卡数为ρ且为森梦想曲面的雅可比椭圆曲面。
英文摘要
Let $π\colon X\to\mathbb P^1$ be a 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 $I_{n_1},\ldots,I_{n_s}$. We prove that the zero section and the components of the reducible fibers generate the closed Mori cone if and only if \[ \sum_{i=1}^s\frac{\lfloor n_i^2/4\rfloor}{n_i}\leχ. \] If $\sum_i n_i\le2χ+3$, then $X$ is a Mori dream surface. The proof combines an explicit description of the facets of the cone generated by the curves visible in the fibration with Artin's criterion applied to the null loci of the dual nef rays. We also show that, in Kodaira dimension one, finiteness of both the Mordell--Weil group and the automorphism group does not mply polyhedrality of the Mori cone. In the polyhedral range, we construct a Jacobian elliptic surface with $(χ,n)=(3,11)$, Picard number $12$, and a big and nef divisor which is not semiample; in particular, this surface is not a Mori dream surface. Finally, for every integer $ρ\ge2$, we construct a Jacobian elliptic surface of Kodaira dimension one and Picard number $ρ$ that is a Mori dream surface.
Comments26 pages, revised version, generalizing the original Theorem 1.1 to semistable fibrations, comments are welcome!
