Intersection Multiplicities

这项建议涉及与交集的重数和除数类群的研究有关的问题。研究人员研究的交集重数包括Serre重数、Duta极限重数和Hilbert-kunz重数，所有这些都是密切相关的，最引人注目的是通过Baum、Fulton和MacPherson的局部化陈角色理论。特别值得注意的是：特征p和特征零情形之间的联系，当两个模都具有有限的射影维度时，Serre的重数可能的不当行为，以及ToR的刚性问题。其中一些项目将与阿努拉格·辛格合作。第二个项目涉及研究除数类群如何在一个簇和该簇中的超曲面之间变化。第三个项目涉及确定对称混合阶梯行列式变体的除数类群。交换代数是发展代数几何基础的重要工具。与数论一起，这些领域是关于经典问题的革命性新方法和思维方式的中心，这些问题最终导致了上个世纪中叶许多数学的现代化。诺特、克鲁尔、韦尔、格罗森迪克、塞雷和扎里斯基都是带来这场革命的人。尤其是交叉理论的现代发展，它仍然是一个特别令人兴奋和重要的研究领域。在同一时期，人们认识到了除子类群的重要性，从那时起，簇中的除子类群与簇中的超曲面之间的联系一直是人们感兴趣的问题。最终，这些研究导致了应用，例如最近在编码理论中的代数几何码的例子。

This proposal concerns questions related to intersection multiplicities and the study of divisor class groups. The intersection multiplicities studied by the investigator include Serre's multiplicity, Dutta's limit multiplicity and the Hilbert-Kunz multiplicity, all intimately related, most notably via the theory of localized Chern characters of Baum, Fulton, and MacPherson. Of particular interest are the following: the connection between characteristic p and characteristic zero situations, the possible misbehavior of Serre's multiplicity when both modules have finite projective dimension, and the question of the rigidity of Tor. Some of these projects will be joint work with Anurag Singh. The second project involves the study of how the divisor class group changes between a variety and a hypersurface inside that variety. The third project involves determining the divisor class groups of symmetric mixed ladder determinantal varieties.Commutative algebra is a crucial tool in developing the foundations of algebraic geometry. Together with number theory, these fields were in the center of the revolutionary new approaches and ways of thinking about classical problems which were eventually responsible for the modernization of much of mathematics in the middle of this last century. Noether, Krull, Weil, Grothendieck, Serre and Zariski were among those who brought about this revolution. Especially central was the modern development of intersection theory and it continues to be an especially exciting and important area of research. In the same time period, the importance of divisor class groups was realized and the connection between those of a variety and a hypersurface in the variety has been a question of interest since then. Eventually these studies lead to applications, such as the recent example of algebro-geometric codes in coding theory.