Intersection Theory for Non Intersectional Cycles

在这个项目中，研究人员研究了两个循环的交集，这些循环的维度加起来不等于环境空间的维度。在以前的交集理论研究中，这是被忽略的情况，因为通常这些循环不相交(因此我们称之为非相交循环)。圈的空间对我们来说是一个很大的谜题，而这个不相交的情况是这个谜题中仍然缺失的一块。因此，为了对周期的空间有一个完整的图景，我们还应该包括不相交的周期。调查者研究的第一个案例是连接案例，其中循环的维度加起来等于比环境空间的维度小一的数字。基本的方法是，这样的交集理论不仅应该包括相遇的周期，也应该包括不相交的周期。为了实现这一点，研究者从Arakelov几何中借用了一个工具-阿基米德高度配对(或者更准确地说，是由Gillet和Soule开发的算术交集理论)，该工具仅为不相交的链环对定义。在这个方向上，这位研究者取得了重大进展：(1)他得到了阿基米德高度对的渐近性的先导项公式。(2)研究了马祖尔的关联结构，构建了周氏品种的关联因子。(3)在上述两个结果的基础上，证明了Clemens猜想：一般五次三重曲线只能容纳有限多条每次光滑有理曲线。该计划是为了加深对包括一般非交叉循环在内的交集理论的理解。主要内容包括：(1)关联因子的研究；(2)关联等价与Abel-Jacobi等价之间的关系；(3)Chow群与Chow簇之间关系的研究；(4)Chow群上Beilinson-Bloch滤子构造的应用。数学中最基本的问题之一是代数方程的求解。一旦人们意识到不能总是明确地写下方程的解，范式就变成了考察不同类型问题的模式，例如：解是否存在，如果存在，有多少解，解集合是否有附加结构？这些都是代数几何中的基本问题。为了回答这些问题，数学家们发展了各种技术，其中之一--交集理论--研究两个或多个方程组的解集的交集。在这个项目中，研究人员计划开发一种新的相交理论方法。这个项目的意义在于研究目前交叉点理论技术较少研究或完全未触及的材料。

In this project the investigator studies the intersection of two cycles whose dimensions do not add up to the dimension of the ambient space. In previous studies of intersection theory this is the case that has been ignored, since in general these cycles do not meet (thus we call them non-intersectional cycles). The space of cycles is a big puzzle to us, and this non-intersectional case is a piece that is still missing from the puzzle. So in order to have a complete picture of space of cycles, one should also include non-intersectional cycles. The first case studied by the investigator was the linking case where the dimensions of cycles add up to the number that is one less than the dimension of the ambient space. The fundamental approach is that such an intersection theory should not only include the cycles that meet, but also those that do not meet. To accomplish this the investigator borrows a tool-Archimedean height pairing from Arakelov geometry (or to be precise, the Arithmetic intersection theory developed by Gillet and Soule), which is only defined for pairs of linking cycles that do not meet. In this direction the investigator has made significant progress: (1) He obtained formulas for the leading term of the asymptotics of Archimedean height pairing. (2) Investigating Mazur's incidence structure, he constructed an incidence divisor on the Chow variety. (3) Based on above two results, he gave a proof of Clemens' conjecture: generic quintic three folds admit only finitely many smooth rational curves of each degree. The plan is to further the understanding of this intersection theory that includes general non-intersectional cycles. The project is concentrated in (1) the study of incidence divisors, (2) the relation between the incidence equivalence and the Abel-Jacobi equivalence, (3) the application to a study of the relation between the Chow group and the Chow variety, (4) the application to a construction of Beilinson-Bloch filtration on the Chow group. One of the most fundamental problems in mathematics is the solving of algebraic equations. Once people it was realized that one could not always explicitly write down the solutions of equations, the paradigm changed into the mode of examining different types of questions such as: does a solution exist, if so, how many solutions are there, do the solution sets have additional structure? These are the fundamental questions in algebraic geometry. In order to answer them, mathematicians have developed varied techniques, one of which-intersection theory--studies he intersection of solution sets of two or more systems of equations. In this project, the investigator plans to develop a new method in intersection theory. The significance of this project is to investigate material that is less studied or completely untouched by the current techniques of intersection theory.