Topics in algebraic geometry

本研究主要围绕代数簇上的主丛展开。在学习丛时，涉及到独立感兴趣的切线主题，如派生范畴、K-理论和Gromov-Witten不变量。这些主题已经在数学物理中得到了应用，我们打算对这篇文献做出贡献，并以此为指导。*这项研究系统地使用了代数堆栈。这些是添加了对称性的几何对象。它们的优点在于，许多问题的自然解都是代数堆栈而不是空间。堆栈在工作中表现为参数空间和具有附加结构的曲线。*我们使用的下一个工具是派生类别及其各种增强。派生范畴是一种具体化的对象。它把一个复杂的对象替换为更容易访问的对象，同时封装了它的所有重要不变量和性质。*使用这些工具，我们旨在研究三个基本问题。*第一个问题是数学家基本感兴趣的。一般理论认为，某些曲线和映射可以通过系数在有理数的有限范围内的方程来定义。然而，对于具体是哪种扩展，人们知之甚少。换句话说，我们有一个存在定理，我们希望使其具有建设性。*第二个问题是关于计算特定数字的面积。我们想要计算固定抛物丛的极大子丛。我们打算使用的方法将利用源自数学物理的思想。我们打算在计算过程中推广一些重要的组合公式。*最后要解决的问题是关于参数的计算。我们有一个重要的普遍对象，主丛的模堆栈，它出现在数学和物理的许多部分。关于它可以问的一个基本问题是，一般情况下，有多少参数取决于。鉴于对象的普遍存在，这是一个应该解决的重要问题。**

This research is centered around principal bundles on algebraic varieties. In studying bundles, tangential subjects of independent interest, such as derived categories, K-theory and Gromov-Witten invariants are touched upon. These subjects have found applications in mathematical physics and we intend to both contribute to and be guided by this literature.*** This research makes systematic use of algebraic stacks. These are geometric objects with added symmetries. Their virtue lies in that many problems have their natural solutions as algebraic stacks rather than spaces. Stacks show up in the work as parameter spaces and as curves with added structure.*** The next tool that we make use of is the derived category and its various enhancements. The derived category is a kind of abelianizing object. It takes a complicated object and replaces it with something more accessible which at the same time encapsulates all its important invariants and properties.***Using these tools we aim to study three fundamental questions.***The first question is of basic interest to number theorists. General theory says that certain curves and maps can be defined via equations that have coefficients in a finite extension of the rational numbers. However not much is known about which particular extension. In other words, we have an existential theorem that we wish to make constructive. ***The second question is about counting are calculation of a particular number. We wish to count maximal subbundles of a fixed parabolic bundle. The approach that we intend to use will make use of ideas originating in mathematical physics. We intend to generalise some important combinatorial formulas in the process of the calculation.***The last question that will be addressed is about calculating parameters. We have an important universal object, the moduli stack of principal bundles that appears in many parts of mathematics and physics. A basic question that can be asked about it is, how many parameters does in generally depend on. Given the objects ubiquity, this is an important question that should be addressed.**