ゴナリティー型不変量と3次元代数ファイバー空間のスロープ不等式

Abstract

代数ファイバー空間には、スロープと呼ばれる数値不変量が定まる。ファイバーの幾何学的性質とスロープの関係を明らかにすることは、代数ファイバー空間の地誌学における主要な問題の一つである。代数ファイバー空間の地誌学的観点から、3次元代数ファイバー空間に対して成立する新たなスロープ不等式を紹介する。

For algebraic fibered spaces, a numerical invariant called the slope is defined. In the study of the geography of algebraic fibered spaces, one of the fundamental problems is to clarify the relationship between the geometric properties of the fibers and the lower bound of the slope. For two-dimensional algebraic fibered spaces, i.e., fibered surfaces, slope inequalities have been established for various geometric properties. On the other hand, slope inequalities for fibered 3-folds are less well-known. We focus on slope inequalities in fibered 3-folds. In this study, two birational invariants are focused on. One is an invariant called the covering gonality. The covering gonality is the higher-dimensional generalization of gonality, the classical invariant of projective curves. The other is an invariant newly introduced, called the minimal covering degree. In the talk, I will explain the relationship between the geometric properties of the fibers captured by these two invariants and the slope inequalities of fibered 3-folds.