Projective embeddings of algebraic varieties

在牛津大学攻读博士学位期间，我将从事代数几何领域的研究。这个数学领域研究被称为代数变量的对象的性质：由多项式方程定义的几何形状。例如,方程y = x2, y2 = x2都来自二级多项式y - x2和y 2 x2,然而他们描述不同平面的几何对象:前者产生了一个经典的“U”形图(一个不可约,抛物线,不可约多项式的定义),而后者提供了一个“X”形状(可约联盟的两条直线方程y = X和y = - X,通过保理的定义多项式)。可约性或其他性质只是一个例子，我们可以赋予一个给定的代数种类，以便对它们进行分类。数学中许多自然发生的物体都可以用一组多项式来描述，但涉及的变量和方程的数量可能非常大。所得到的几何对象可能难以明确地描述。然而，我们可以使用代数几何中的方法，在计算机代数包（如Macaulay2或Singular）的帮助下，建立这些对象的属性：它们的维度，是奇异的还是非奇异的，等等。常见的例子包括某些向量空间（Grassmannian和flag变体）的给定维数的线性子空间集合，或者具有一定数量标记点的g属曲线的模空间，这是我在早期的夏季项目中研究过的。代数集在数学中的普遍存在性赋予了代数几何广泛的应用。在机器人技术中，定义不同部件相对运动的方程可以归结为一个多项式方程系统。现代密码学使用椭圆曲线，这是代数变种的例子。在理论物理中，弦理论家对代数几何也很感兴趣，空间本身被认为是10维的，其中有一个6维的“分量”，这是一种特殊的代数变体。我的工作将旨在利用Miles Reid首创的分级环方法，结合计算机代数方法，在某些特定环境下理解代数变体的投影嵌入。前面提到的曲线的模空间，以及一些被称为希尔伯特点格式的代数曲面上的模空间，将是我们特别感兴趣的焦点。该项目属于EPSRC代数研究领域，将由Balazs Szendroi教授指导。

During my DPhil studies in Oxford, I will be working in the field of Algebraic Geometry. This field of mathematics studies properties of objects called algebraic varieties: geometric shapes defined by polynomial equations. For instance, the equations y = x2 and y2 = x2 both come from second degree polynomials y - x2 and y 2 - x2 , yet they describe very different geometric objects in the plane: the former yields a classic 'U' shaped graph (an irreducible variety, the parabola, defined by an irreducible polynomial), while the latter gives an 'X' shape (a reducible union of two lines with equations y = x and y = -x , obtained by factoring the defining polynomial). Reducibility or otherwise is just one example of a property we can assign to a given algebraic variety in order to classify them.Many naturally occurring objects in mathematics can be described by a set of polynomials, but the number of variables and equations involved may be very large. The resulting geometric objects can be difficult to describe explicitly. However, we can use methods in algebraic geometry, aided by computer algebra packages such as Macaulay2 or Singular, to establish properties of these objects: their dimension, being singular or otherwise, and others. Popular examples include sets of linear subspaces of given dimension of some vector space (Grassmannian and flag varieties), or the moduli space of curves of genus g with some number of marked points, which I studied in an earlier summer project.The ubiquitous nature of algebraic sets across mathematics gives algebraic geometry a broad array of applications. In robotics, the equations defining movement of different parts relative to one another boils down to a system of polynomial equations. Modern cryptography uses elliptic curves, which are examples of algebraic varieties. Algebraic geometry is also of interest to string theorists in theoretical physics, where space itself is considered as 10-dimensional, with a 6-dimensional 'component' a special kind of algebraic variety.My work will aim to understand projective embeddings of algebraic varieties in some specific contexts, using the Graded Ring method pioneered by Miles Reid, combined with computer algebra methods. Moduli spaces of curves mentioned before, as well as certain moduli spaces attached to algebraic surfaces called Hilbert schemes of points, will be a particular focus of interest. This project falls within the EPSRC Research Area Algebra, and will be supervised by Professor Balazs Szendroi.