Arakelov geometry and its related topics

本课题的主要结果是（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)考虑了交换簇上的问题，证明了Lang猜想和Bogomolov猜想的一个推广。设K是Q上的一个生成域，A是K上的一个交换簇。然后，使用一个好的高度函数，我们可以定义一个高度配对<，>：A（F）× A（F）-> R，这是一个数域上的Neron-Tate高度配对的扩展（注意F是K的代数闭包）。对于x_1，.，x_r ∈A（F），我们将det（<x_i，x_j>）记为δ（x_1，.，x_r）。设Γ是A（F）的有限秩子群，X是A的子簇.，x_n}是Γ的Q-基。然后，我们得到，如果集合{ x ∈ X（F）|δ（x_1，.，x_n，x）<$ε }在X中是Zerkiki稠密的，则X是交换子簇被Γ_ {div}中的一个元素平移的结果. (2)设X是正规完备簇，U是X的Zapriki开集，称X上的Q-线丛L在U上nef，如果对通过U的任意完备曲线C，L与C的交数非负.设M_g是亏格g的稳定曲线的模空间，M_g^1是由最多只有一个结点的稳定曲线组成的Zapriki开集。然后，利用M_g上的重言式类，我们确定了M_g上的Q-线丛是nef over M_g^1的充要条件。利用这一点，我们可以看出由通过M_g^1的曲线生成的圆锥是一个有理多面体，并且我们还可以具体地描述圆锥的极值射线。我们认为这是迈向关于M_g，n的Mori锥的富尔顿猜想的第一步。

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.