Moduli Spaces for Rings and Ideals

研究者研究了有限平面覆盖和这些覆盖的线束的模空间。重要的基本例子包括数域的阶（整数的有限平面覆盖）和复射影线的有限覆盖。这些方法涉及到在任意基方案上工作，因此，例如，一个人可以得到关于上述数论和几何例子的结果。该项目是寻找具有明确描述和合理几何的模空间，以便可以具体地处理它们。自1801年高斯的工作以来，多项式已被用于研究比通常计数数1、2、3更大的数字系统。例如，一个更大的数字系统可能还包括根号2，它在1、2、3……中找不到。当我们包括根号2时，它是通常数字的二次展开式，如果我们包括根号2，它就是通常数字的三次展开式。这项工作试图通过明确地处理与扩展相关的多项式来理解通常数字的可能的低次扩展是什么。这就允许人们通过将它们简化为多项式的简单计算来对更大的数字系统进行计算。

The investigator studies moduli spaces of finite, flat covers and line bundles of those covers. Important basic examples include orders in number fields (finite flat covers of the integers) and finite covers of the complex projective line. The methods involve working over an arbitrary base scheme, so for example one gets results about both the number theoretic and geometric examples above. The project is to find moduli spaces that have explicit descriptions and reasonable geometry so that they can be worked with concretely. Since the work of Gauss in 1801, polynomials have been use to study number systems that are bigger than the usual counting numbers 1,2,3.., For example, a larger number system might also include the square root of 2, which cannot be found among 1,2,3... When we include the square root of 2, it is a quadratic extension of the usual numbers, and if we had included the cube root of 2 it would have been a cubic extension of the usual numbers. This work tries to understand what the possible low degree extensions of the usual numbers are by working explicitly with polynomials that are related to the extensions. This then allows one to make computations regarding the larger number systems by reducing them to easier computations about polynomials.