标题： 整数点和射影平面上自同态的轨道 显示英文标题

标题： Integral points and orbits of endomorphisms on the projective plane

摘要： 我们分析在$\mathbb{P}^2$中有限多个曲线的补集上的整点何时可能是稠密的。 我们根据其对数Kodaira维数$\bar{\kappa}$来对这些仿射曲面进行分析。 当$\bar{\kappa} = -\infty$时，我们完全通过表面上无穷远处不可约分支的数量以及与曲面自然相关的线性系中的多重成员的数量来描述整点的潜在稠密性。 当整点不是潜在稠密时，我们证明它们位于有限多个可以有效计算的曲线上。 当$\bar{\kappa} = 0$时，我们证明整点总是潜在稠密的。 我们的分析大部分集中在$\bar{\kappa}=1$的微妙情况下。 我们在若干情况下确定了整点的潜在稠密性，并开发了用于研究纤维化到曲线上的曲面上整点的工具。 最后，在$\bar{\kappa}=2$的情况下，整点的非稠密性由Lang-Vojta猜想预测，对此我们没有新的内容可以补充。 在相关方向上，我们研究了在$\mathbb{P}^2$的自同态作用下的轨道中的整点。 假设朗-沃贾塔猜想，我们证明了一个自同态$\phi$在$\mathbb{P}^2$上的轨道中，只有在存在一个相对于$\phi$的非平凡完全不变的真Zariski闭集的情况下，才能包含一个相对于某个非平凡有效除子的Zariski稠密的整点集。 这可以看作是Silverman关于有理函数轨道中整点结果的一个推广。 我们提供了许多具体的例子，并以一些开放问题结束。 显示英文摘要

摘要： We analyze when integral points on the complement of a finite union of curves in $\mathbb{P}^2$ are potentially dense. We divide the analysis of these affine surfaces based on their logarithmic Kodaira dimension $\bar{\kappa}$. When $\bar{\kappa} = -\infty$, we completely characterize the potential density of integral points in terms of the number of irreducible components on the surface at infinity and the number of multiple members in a pencil naturally associated to the surface. When integral points are not potentially dense, we show that they lie on finitely many effectively computable curves. When $\bar{\kappa} = 0$, we prove that integral points are always potentially dense. The bulk of our analysis concerns the subtle case of $\bar{\kappa}=1$. We determine the potential density of integral points in a number of cases and develop tools for studying integral points on surfaces fibered over a curve. Finally, nondensity of integral points in the case $\bar{\kappa}=2$ is predicted by the Lang-Vojta conjecture, to which we have nothing new to add. In a related direction, we study integral points in orbits under endomorphisms of $\mathbb{P}^2$. Assuming the Lang--Vojta conjecture, we prove that an orbit under an endomorphism $\phi$ of $\mathbb{P}^2$ can contain a Zariski-dense set of integral points (with respect to some nontrivial effective divisor) only if there is a nontrivial completely invariant proper Zariski-closed set with respect to $\phi$. This may be viewed as a generalization of a result of Silverman on integral points in orbits of rational functions. We provide many specific examples, and end with some open problems.