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束模堆栈的基本维度。几何对象的尺寸是一个重要的不变量。堆栈是普通空间的重要概括，在考虑对象的参数空间时自然出现。例如，曲线最自然的参数“空间”实际上不是空间而是堆栈。人们可能将堆栈视为一种现代对称理论。对于堆栈有两个维度概念。经典尺寸可以通过几十年前开发的变形理论的方法来计算。维度的第二个概念，本质维度，被证明是一个更加微妙的不变量。它是大量密集工作和有趣猜想的中心。在我们的研究中，我们有兴趣研究代数曲线上某些堆栈参数化丛的基本维度。 这项工作的第二个主题是研究射影线减三点上特征零的 Nori 有限抛物线丛的范畴。在数学中，物体的对称性集合称为群。人们可以考虑具有某些属性的数字对称性的集合，或者更准确地说是有理数的绝对伽罗瓦群。这个群很大，很有趣，不幸的是，我们无法对其进行具体描述。在 0 和 1 处穿刺的线的代数覆盖集合具有该群的非常有趣的作用。我们想尝试阐明这一行动。 覆盖对应于两个字母上的自由组的子组，并且可以通过穿孔线的通用覆盖来研究。通用封面是所有封面的始祖。通用覆盖的问题在于，因为它本质上是拓扑的，所以很难描述其有限商上的伽罗瓦作用。我们继续引入通用覆盖的代理，称为线上的 Nori 有限抛物线束类别。这方面的伽罗瓦群作用更加透明，但这一类别尚未描述。描述这一类别是本研究最直接的目标。