Topics in algebraic bundles

代数丛中的主题

• 批准号: 327639-2011
• 负责人: Dhillon, Ajneet
• 金额: $ 1.09万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572082
• 关键词: Topics; algebraic; bundles

The first topic to be considered is the essential dimension of moduli stacks of bundles. The dimension of a geometrical object is an important invariant. Stacks are an important generalisation of ordinary spaces that arise naturally when considering parameter spaces for objects. For example the most natural parameter "space" for curves is in fact not a space but a stack. One may view stacks as a modern theory of symmetries. For stacks there are two notions of dimension. Classical dimension, can be computed via methods from deformation theory that were developed many decades ago. The second notion of dimension, essential dimension, turns out to be a far more subtle invariant. It is the center of much intensive work and interesting conjectures. In our research we are interested in studying the essential dimension of certain stacks parametrising bundles over algebraic curves. The second topic in this work is to study the category of Nori finite parabolic bundles in characteristic zero on the projective line minus three points. In mathematics, the collection of symmetries of an object is called a group. One can consider the collection of symmetries of numbers with certain properties, or more precisely the absolute Galois group of the rational numbers. This group is large, interesting and unfortunately a concrete description of it is not within our grasp.The collection of algebraic covers of the line punctured at 0 and 1 has a very interesting action of this group. We would like to try and shed some light on this action. Covers correspond to subgroups of the free group on two letters and can be studied by the universal cover of the punctured line. The universal cover is the ancestor of all covers. The problem with the universal cover is that because it is topological in nature it is difficult to describe the Galois action on its finite quotients. We proceed by introducing a proxy for the universal cover called the category of Nori finite parabolic bundles on the line. The Galois group action on this side is more transparent but this category has yet to be described. Describing this category is the most immediate goal of this research.

要考虑的第一个主题是基本尺寸的模栈束。几何对象的维数是一个重要的不变量。栈是普通空间的一个重要推广，当考虑对象的参数空间时自然出现。例如，曲线的最自然的参数“空间”实际上不是空间，而是堆栈。人们可以把堆栈看作是对称性的现代理论。对于栈，有两种维数的概念。经典尺寸，可以通过几十年前发展的变形理论的方法来计算。维数的第二个概念，本质维数，是一个更微妙的不变量。它是许多密集的工作和有趣的展览的中心。在我们的研究中，我们感兴趣的是研究代数曲线上的某些栈参数化丛的本质维数。 本文的第二个主题是研究负三点射影线上特征为零的Nori有限抛物丛范畴。在数学中，对象的对称性的集合被称为群。人们可以考虑具有某些性质的数的对称性的集合，或者更精确地说，有理数的绝对伽罗瓦群。这个群体是大的，有趣的，不幸的是，它的具体描述是不是在我们的把握。收集代数覆盖的线穿孔在0和1有一个非常有趣的行动，这个群体。我们想试着解释一下这一行动。 覆盖对应于两个字母上的自由群的子群，并且可以通过穿孔线的泛覆盖来研究。通用封面是所有封面的祖先。泛覆盖的问题在于，因为它本质上是拓扑的，所以很难描述有限子上的伽罗瓦作用。我们继续通过引入一个代理的普遍覆盖称为范畴的Nori有限抛物丛的线。这一边的伽罗瓦群作用更透明，但这一类别还有待描述。描述这一类别是本研究最直接的目标。