Mathematical Sciences: Topics in Algebraic Geometry

Ein Ein建议研究代数几何中的下列问题和相关问题。(1)高维簇上的线性系统。研究代数簇几何的基本工具之一是研究代数簇上线性系统的性质。Ein(与R.Lazarsfeld一起)证明了Fujita关于一大类三重共轭线性系统的自由性的猜想。他们希望应用研究这一猜想的新技术来研究奇异因子和特殊亚簇的几何。特别是，他们计划研究阿贝尔变种上的theta因子的奇异性。他们还希望将他们关于线性系统的结果推广到更高的维度，并研究非常广泛的问题。(2)充分线丛和Seshadri常数的局部正性。Seshadri常数是一个重要的不变量，用来度量充裕线丛的局部正性。Ein想要研究这样的猜想，即对于我们的射影变体中的一般点x，Seshadri常数大于或等于1。(3)代数簇的合子。当簇被一个伴随的线性系统嵌入时，Ein想要研究簇的几何和簇的配位环的合子之间的关系。(4)属完全交叉亚种。Ein证明了高次一般完全交的所有子簇都是一般类型的结果。这一结果最近得到了Voisin的改善。当完备交簇是一个Calabai-Yau流形时，Ein想把这个结果推广到研究一般完全交簇的子簇的变形。这些问题具有新的意义，因为它们与物理学和镜像对称性有关。这是在代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的25年里，它已经取得了革命性的成就。在起源中，它处理的是可以在PLNE中通过最简单的方程定义的图形，即多项式。如今，这个领域不仅使用代数的方法，而且使用分析和拓扑学的方法，相反，在这些领域以及物理、理论计算机科学和机器人学中都有应用。

EIN Ein proposes to investigate the following and related problems in algebraic geometry. (1) Linear systems on higher dimensional varieties. One of the basic tools in investigating the geometry of an algebraic variety is to study the properties of the linear systems on the variety. Ein (jointly with R. Lazarsfeld) proved a conjecture of Fujita on the freeness of a large class of adjoint linear systems on threefolds. They would like to apply the new techniques developed in studying this conjecture to study singular divisors and the geometry of the special subvarieties. In particular they plan to study the singularities of the theta divisors on abelian varieties. They would also like to extend their results on linear systems to higher dimensions and to study questions on very ampleness. (2) Local positivity of ample line bundles and Seshadri constants. The Seshadri constant is an important invariant that measures the local positivity of an ample line bundle. Ein would like to work on the conjecture that the Seshadri constant is greater or equal to one for a general point x in our projective variety. (3) Syzygies of algebraic varieties. Ein would like to investigate the relation between the geometry of the variety and the syzygies of the coordinate ring of the variety while the variety is embedded by an adjoint linear system. (4) Subvarieties od generic complete intersections. Ein has proved a result that all the subvarieties of a generic complete intersection of high degree are of general type. This result has recently been improved by Voisin. Ein would like to extend the result to study the deformations of subvarieties of a generic complete intersection while the complete intersection variety is a Calabai-Yau manifold. These problems take on new significance because of their relations to physics and mirror symmetry. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In the origin, it treated figures that could be defined in the plne by the simplest equations, namesly polynomials. Nowadays, the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding applicaton in those fields as well as in physics, theoretical computer science, and robotics.