Topics in algebraic bundles

要考虑的第一个主题是丛的模堆栈的基本维度。几何物体的尺寸是一个重要的不变量。堆栈是在考虑对象的参数空间时自然产生的普通空间的重要推广。例如，曲线最自然的参数“空间”实际上不是空间，而是堆栈。人们可以将堆栈视为一种现代的对称理论。对于堆栈来说，有两个维度的概念。经典尺寸，可以通过几十年前发展起来的变形理论的方法来计算。维度的第二个概念，本质维度，被证明是一个更微妙的不变量。它是许多密集工作和有趣猜测的中心。在我们的研究中，我们感兴趣地研究了代数曲线上某些栈参数升丛的本质维度。 本文的第二个主题是研究射影直线上特征零减三点的Nori有限抛物丛范畴。在数学中，一个对象的对称性的集合称为群。人们可以考虑具有某些性质的数的对称性的集合，或者更准确地说，是有理数的绝对伽罗瓦群。这个群很大，很有趣，但不幸的是，我们无法掌握它的具体描述。在0和1处穿孔的直线的代数覆盖集合具有这个群的一个非常有趣的作用。我们想试着对这一行动做出一些解释。覆盖对应于两个字母上的自由组的子群，并且可以通过穿孔线的通用覆盖来研究。万能封面是一切封面的鼻祖。万能覆盖的问题是，因为它本质上是拓扑的，所以很难描述它在有限商上的伽罗瓦作用。接着，我们在直线上引入了泛盖的一个代理，称为Nori有限抛物丛范畴。这一方的伽罗瓦集团行动更加透明，但这一类别尚未描述。描述这一范畴是本研究最直接的目标。

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.