Mathematical Sciences: Number of Defining Equations, Local Cohomology

本课题研究了局部上同调、线性上同调和代数变量的定义方程数等相关问题。最近在这些问题上取得了一些进展，但仍有许多工作要做。仿射空间的每一个非奇异子变是否都是集合论的完全交，目前尚不清楚。局部上同模的结构仍然是一个谜。上同维数与上同维数之间的关系尚不清楚。关于局部传说上同的一些技术问题仍未解决。首席研究员将使用过去成功的方法以及开发一些新方法来研究这些问题。代数几何是研究由多项式方程组的零点集合所产生的几何对象的学科。这是数学中最古老也是目前最活跃的分支之一。它在数学、计算机科学和物理学中有着广泛的应用。

This project is concerned with the study of some interrelated questions of local cohomology, etale cohomology and the number of defining equations of algebraic varieties. Some progress has been made on these problems recently, but much remains to be done. It is still unknown whether every non-singular subvariety of affine space is a set-theoretic complete intersection. The structure of local cohomology modules remains a mystery. The relationship between the cohomological dimension and the etale cohomological dimension is still unclear. Some technical questions about local etale cohomology remain open. The principal investigator will study these questions using methods that have been successful in the past as well as developing some new methods. Algebraic geometry is the study of the geometric objects arising from the sets of zeros of systems of polynomial equations. This is one of the oldest and currently one of the most active branches of mathematics. It has widespread applications in mathematics, computer science and physics.