Families of Curves on Algebraic Surfaces

研究者将研究平面曲线的几何，特别是具有给定度和属并满足附加几何条件的平面曲线的Severi变种。这将从三个方向进行。首先，研究者将扩展现有的关于Severi品种的不可约性和程度的工作，以参数化具有节点以外的奇异点的平面曲线。其次，他将应用最近发展的技术来描述这些变体的某些超平面截面，并可能应用于有关这些变体的皮卡德群及其上的通用曲线的问题（如恩里克猜想）。最后，研究者将尝试使用相同的思想来分析一个相关的问题，即在指定的一般点上具有奇点的平面曲线线性序列的维数。这是代数几何领域的研究。代数几何处理的是非常基本的物体，可以用最简单的方程（即多项式）在平面和空间中定义的图形。因此，它是现代数学中最古老的部分之一——比如古希腊人就研究过它。与此同时，它也是最新的一种，在过去的四分之一世纪里经历了革命性的繁荣。在代数几何中，也许最基本和最彻底的研究课题是代数曲线，即由两个变量的单个多项式方程定义的平面上的曲线。然而，尽管人们对这些天体做了很多经典的研究，我们对它们的理解仍在发展，而且速度很快；这是一个非常活跃的调查领域。

Harris The investigator will study the geometry of plane curves, and specifically the Severi varieties papametrizing plane curves having given degree and genus and satisfying additional geometric conditions. This will proceed in three directions. First, the investigator will extend existing work on irreducibility and degree of Severi varieties to varieties parametrizing plane curves with singularities other than nodes. Second, he will apply recently developed techniques to describe certain hyperplane sections of these varieties, with possible applications to questions (such as the Enriques conjecture) concerning the Picard groups of these varieties and of the universal curves over them. Finally, the investigator will attempt to use the same ideas to analyze a related question, that of the dimension of the linear series of plane curves with singularities at assigned general points. This is research in the field of algebraic geometry. Algebraic geometry deals with very basic objects, figures that can be defined in the plane and in space by the simplest equations, namely polynomialsis. It is thus one of the oldest parts of modern mathematics--it was studied, for example by the ancient Greeks. At the same time it is one of the newest, having had a revolutionary flowering in the past quarter-century. Within algebraic geometry, perhaps the most basic and most thoroughly studied subject is that of algebraic curves, that is, curves in the plane defined by a single polynomial equation in two variables. For all the work that has been done on these objects classically, however, our understanding of them is still developing, and at a rapid pace; it is a very active field of investigation.