Geometric Invariant Theory
Geometric Invariant Theory
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DOI：
10.1090/gsm/184/09
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发表时间：
2011
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影响因子：
--
通讯作者：
Atanas Atanasov
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文献类型：
--
作者：
Atanas Atanasov
介绍几何不变理论是在代数几何的背景下研究同数。许多我们希望取商的对象都具有某种几何结构，几何不变理论（GIT）允许我们构造保持几何结构的商。商是数学中自然产生的对象。给定一个在元素上具有等价关系的对象，商给出了一个更简单的对象，它保留了关于原始对象的信息，同时删除了不必要的数据，通过将等价元素视为相同的。当我们研究一个对象时，我们可能有一个等价的元素，而我们不关心等价元素之间的区别。举例来说，在研究矩阵时也会出现类似的情况--我们经常想研究矩阵的相似类，而不区分相似矩阵。我们让一个群G作用在一个几何对象X上。G的作用给出了X在G-轨道上的一个划分，它定义了X上的一个等价关系。G-轨道的集合并不总是具有几何结构。几何不变理论商是一种在一定程度上划分G-轨道的构造，同时保留一些理想的几何性质和结构。对于仿射集，GIT商的构造是很好理解的，并且唯一确定。在投影的情况下，构造商的自然方法是将仿射集的向量粘合在一起。这不能总是简单地或以独特的方式完成。G-作用必须推广到正确覆盖仿射子集，这在构造中引入了一定程度的选择。一个显而易见的问题是：商是如何根据这种选择构建的？更进一步：在不同构造的替代物之间存在任何关系吗？我将建立GIT商数的结构，并探讨以不同方式构建的商数之间存在的关系。在第一章中，我将介绍一些代数几何的背景知识。我将介绍一个品种的概念，研究的对象，和拓扑结构上使用的品种。我将介绍代数和几何之间的关系，在这种情况下，以及如何一个品种可以确定其坐标环，在仿射和射影的情况下。第二章介绍了簇范畴中的态射。在代数几何中，我们主要关注的是一组函数.
Introduction Geometric Invariant Theory is the study of quotients in the context of algebraic geometry. Many objects we would wish to take a quotient of have some sort of geometric structure and Geometric Invariant Theory (GIT) allows us to construct quotients that preserve geometric structure. Quotients are naturally arising objects in mathematics. Given an object with an equivalence relation on the elements, the quotient gives a simpler object which retains information about the original object whilst removing unnecessary data, by considering equivalent elements as the same. When we study an object, we may have an equivalence of elements where we are not concerned with the distinction between equivalent elements. By way of example, an analogous situation areises when studying matrices-we often want to study the similarity classes of matrices and not distinguish similar matrices. We let a group G act on a geometric object X. The action of G gives a partition of X in to G-orbits, which defines an equivalence relation on X. It is not always the case that the set of G-orbits has a geometric structure. The Geometric Invariant Theory quotient is a construction that partitions G-orbits to some extent, while preserving some desirable geometric properties and structure. For affine sets, the construction of the GIT quotient is well understood and is determined uniquely. In the projective case, the natural way to construct a quotient is to glue together quotients of affine sets. This can not always be done simply, or in a unique way. The G-action must be extended to cover affine subsets correctly, which introduces a certain degree of choice in the construction. An obvious question to ask is: How is the quotient constructed dependent on this choice? And further: Are there any relations that exist between differently constructed quotients? I will set up the construction of the GIT quotient and explore the relations existing between quotients constructed in different ways. In the first chapter, I will give some background on algebraic geometry. I will introduce the concept of a variety, the objects of study, and the topology used on varieties. I will introduce the relation between algebra and geometry in this setting, and how a variety can be identified with its coordinate ring, in both the affine and projective case. i The second chapter introduces the morphisms in the category of varieties. In algebraic geometry we are largely concerned with the set of functions …