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叉点，它们的维度加起来不等于周围空间的维度。 在以前的交叉理论研究中，这种情况被忽略了，因为一般来说这些循环并不相交（因此我们称之为非交叉循环）。 循环的空间对我们来说是一个大难题，而这个非交叉的例子是这个难题中仍然缺少的一块。 因此，为了对循环空间有一个完整的了解，我们还应该包括非交叉循环。 研究者研究的第一种情况是连接情况，其中循环的维数加起来等于比周围空间的维数小1的数。 基本的方法是，这样的交集理论不仅应该包括满足的循环，而且还应该包括不满足的循环。 为了实现这一点，研究者借用了阿拉克洛夫几何（或者更准确地说，由吉莱和索尔开发的算术相交理论）的工具阿基米德高度配对，它只定义为不相交的链接循环对。 在这个方向上，研究者取得了重大进展：（1）他获得了阿基米德高度配对渐近性的首项公式。 (2)调查马祖尔的发病率结构，他建造了一个发病率因子的周品种。 (3)基于上述两个结果，他给出了Clemens猜想的一个证明：一般五次三重曲线只允许每一次的光滑有理曲线的个数为1/2。 该计划是为了进一步了解这个交叉理论，包括一般的非交叉周期。 本课题主要研究了关联因子的研究，关联等价与Abel-Jacobi等价的关系，关联等价在Chow群与Chow簇关系研究中的应用，关联等价在Chow群上的Beilinson-Bloch滤子构造中的应用。数学中最基本的问题之一是解代数方程。 一旦人们意识到一个人不能总是明确地写下方程的解，范式就转变为检查不同类型问题的模式，例如：是否存在解，如果存在，有多少解，解集是否有额外的结构？ 这些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各样的方法，其中之一--交理论--研究两个或多个方程组的解集的交。 在这个项目中，研究人员计划开发一种新的交叉理论方法。 这个项目的意义是调查材料，是较少研究或完全接触到目前的技术相交理论。

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.