Projective embeddings of algebraic varieties

在牛津大学攻读哲学博士期间，我将从事代数几何领域的工作。这一数学领域研究称为代数簇的对象的属性：由多项式方程定义的几何形状。例如，方程y = x2和y2 = x2都来自二次多项式y-x2和y2- x2，但它们描述了平面上非常不同的几何对象：前者产生经典的“U”形图（一个不可约的品种，抛物线，由不可约的多项式定义），而后者给出“X”形状（具有方程y = x和y = -x的两条线的可约并集，通过因式分解定义多项式获得）。约简或其他性质只是我们可以赋予给定代数簇以分类它们的性质的一个例子。数学中许多自然发生的对象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。生成的几何对象可能难以明确描述。然而，我们可以使用代数几何中的方法，借助计算机代数软件包，如Macaulay 2或Singular，来建立这些对象的属性：它们的维度，奇异或其他。流行的例子包括一些向量空间（格拉斯曼和旗簇）的给定维数的线性子空间的集合，或者具有一定数量标记点的亏格g曲线的模空间，我在早些时候的夏季项目中研究了代数集合在数学中的无处不在的性质给代数几何带来了广泛的应用。在机器人技术中，定义不同部件相对于彼此的运动的方程归结为多项式方程组。现代密码学使用椭圆曲线，这是代数簇的例子。代数几何也是弦理论家在理论物理学中感兴趣的，空间本身被认为是10维的，其中6维“分量”是一种特殊的代数簇。我的工作将旨在理解代数簇在某些特定情况下的投影嵌入，使用Miles Reid开创的分次环方法，结合计算机代数方法。前面提到的曲线的模空间，以及某些附加到代数曲面的模空间，称为点的希尔伯特方案，将是一个特别感兴趣的焦点。该项目属于EPSRC研究领域代数的福尔斯，并将由Balazs Szendroi教授监督。

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.