论文标题
在对数案例中的Urata定理和整体上的应用
Urata's theorem in the logarithmic case and applications to integral points
论文作者
论文摘要
乌拉塔（Urata）表明，尖锐的紧凑双曲线品种仅从尖曲线中有限的多个地图。我们将URATA定理扩展到（不一定是紧凑的）夸张的可嵌入品种的设置。作为一个应用程序，我们表明，在数字字段$ k $上只有一个有限的$ \ mathcal {o} _ {o} _ {l，t} $ - 任何数字字段$ l/k $以及任何有限的有限位置的$ t $ l $的$ t $ forneriation Bortialitiation nyteriation for Mathbb filtery nyteraliation for n norme fienter，n数字$ t $ fientraliation for n normuite colteraliatiation n normuite coltiatiation $ firital in fienter-firator，我们将后一个结果与格林的标准结合使用，用于双曲线嵌入性，以获得平滑仿射曲线的对称自我产生的积分点的新颖有限性结果，以及投影品种中大型除数的补充。最后，在曲线的对称自我产生的情况下，我们使用与格林标准的部分交谈来进一步研究双曲线嵌入性（或其失败）。作为我们结果的副产品，我们获得了平滑仿射的brody-hyperbolic三倍以$ \ mathbb {c} $的示例，该示例不是夸张的嵌入。
Urata showed that a pointed compact hyperbolic variety admits only finitely many maps from a pointed curve. We extend Urata's theorem to the setting of (not necessarily compact) hyperbolically embeddable varieties. As an application, we show that a hyperbolically embeddable variety over a number field $K$ with only finitely many $\mathcal{O}_{L,T}$-points for any number field $L/K$ and any finite set of finite places $T$ of $L$ has, in fact, only finitely many points in any given $\mathbb{Z}$-finitely generated integral domain of characteristic zero. We use this latter result in combination with Green's criterion for hyperbolic embeddability to obtain novel finiteness results for integral points on symmetric self-products of smooth affine curves and on complements of large divisors in projective varieties. Finally, we use a partial converse to Green's criterion to further study hyperbolic embeddability (or its failure) in the case of symmetric self-products of curves. As a by-product of our results, we obtain the first example of a smooth affine Brody-hyperbolic threefold over $\mathbb{C}$ which is not hyperbolically embeddable.
