Arakelov geometry and its related topics

本文的主要结果是：(1)有限生成域上有理点的分布，(2)稳定曲线及其锥的模空间的Picard群.(1)我们考虑了交换簇上的问题，证明了Lang猜想和Bogomolov猜想的推广.设K是Q上的有限生成域，A是K上的阿贝尔变种。然后，利用Head调查者提供的一个好的高度函数，我们可以定义一个高度配对&lt；，&gt；：A(F)×A(F)-&gt；R，它是数域上Nelon-Tate高度配对的推广(请注意，F是K的代数闭包)。对于x_1，…，x_r∈A(F)，我们用δ(x_1，...，x_r)表示det(&lt；xi，x_j&gt；)。设Γ是A(F)的有限秩子群，X是A的子簇，{x_1，…，x_n}是Γ的Q-基。然后，如果集合{x∈X(F)|δ(x_1，...，x_n，x)≦ε}在X中是Zariski稠密的，则X是交换子簇由ε_{div}的元素平移而成。(2)设X是正规完备簇，U是X的Zariski开集。设X上的Q-线丛L在U上是NEF的，如果对任何经过U的完备曲线C，L与C的交数都是非负的。设M_g是亏格g的稳定曲线的模空间，M_g^1是由至多一个结点的稳定曲线组成的Zariski开集。然后，根据M_g上的重言式类，我们确定了保证M_g上的Q-线丛在M_g^1上是nef的充要条件。利用这个充要条件，我们可以看到通过M_g^1的曲线所生成的锥体是有理多面体，并且我们还可以用具体的方法描述该锥体的极射线。我们认为这是向Fulton猜想关于M_g，n的Mori锥迈出的第一步。

The main results of this project are (1) Distribution of rational points over a finitely generated field, and (2) The Picard group of the moduli space of stable curves and its cone.(1) We considered the problems over an abelian variety and proved a generalization of Lang conjecture and Bogomolov conjecture. Let K be a finitely generated field over Q and A an abelian variety over K.Then, using a good height function due to the head investigator, we can define a height pairing < , > : A (F) × A (F) -> R, which is an extension of Neron-Tate height pairing over a number field (note that F is the algebraic closure of K). For x_1, ... , x_r ∈A (F), we denote det(<x_i, x_j>) by δ (x_1, ..., x_r) . Let Γ be a finite rank subgroup of A (F), and X a subvariety of A.Moreover, let {x_1, ... , x_n} be a Q-basis of Γ. Then, we obtained that if the set { x ∈ X(F)|δ(x_1, ..., x_n, x)≦ε }is Zariski dense in X for every positive number ε , then X is a translation of an abelian subvariety by an element of Γ_ {div}.(2) Let X be a normal complete variety and U a Zariski open set of X.A Q-line bundle L on X is said to be nef over U if, for any complete curve C passing through U, the intersection number of L with C is non-negative. Let M_g be the moduli space of stable curves of genus g, and M_g^1 the Zariski open set consisting of stable curves with one node at most. Then, in terms of tautological classes on M_g, we determine the necessary and sufficient condition to guarantee that a Q-line bundle on M_g is nef over M_g^1. Using this, we can see that the cone generated by curves passing through M_g^1 is a rational polyhedra, and we can also describe extremal rays of the cone in a concrete way. We believe that this is a first step toward a Fulton conjecture concerning the Mori cone of M_g, n.