Families of Curves on Algebraic Surfaces

哈里斯 研究者将研究平面曲线的几何，特别是具有给定度和亏格并满足附加几何条件的Severi变种papametrizing平面曲线。这将从三个方向进行。首先，研究者将扩展现有的不可约性和度的Severi品种品种的品种parametrizing平面曲线与奇点以外的节点。其次，他将应用最近发展的技术来描述这些品种的某些超平面部分，可能的应用问题（如恩里克斯猜想）有关皮卡德集团的这些品种和普遍的曲线。最后，研究人员将试图使用相同的想法来分析一个相关的问题，即在指定的一般点处具有奇点的平面曲线的线性系列的维数。 这是代数几何领域的研究。代数几何处理非常基本的对象，可以在平面和空间中定义的最简单的方程，即多项式的数字。因此，它是现代数学中最古老的部分之一，例如古希腊人就研究过它。与此同时，它也是最新的一个，在过去的四分之一个世纪里，它经历了一场革命性的繁荣。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，也就是平面上的曲线由一个二元多项式方程定义。然而，尽管人们已经对这些天体进行了大量的经典研究，但我们对它们的理解仍在迅速发展，这是一个非常活跃的研究领域。

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.