| Co-Investigator(Kenkyū-buntansha) |
YOSHIKAWA Ken-ichi Tokyo University., Graduate School of Mathematical Sciences, Associate Professor, 数理科学研究科, 助教授 (20242810)
SAITO Masahiko Kobe University., Faculty of Science Professor, 理学部, 助教授 (80183044)
ISHII Akira Kyoto University., Graduate School of Technology, Lecturer, 大学院・工学研究科, 講師 (10252420)
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| Research Abstract |
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.
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