Linear Systems of Plane Curves With Assigned Singularities

Caporaso 9801465这个项目的目标是解决代数几何中的下列经典猜想：在平面中指定的一般点处具有给定数目的奇点的定次平面曲线的线性系统的维度等于期望值。这样的期望值可以很天真地计算出来，自本世纪初以来，上述问题一直是研究的对象。解决这一问题的工作将使我们更好地理解模空间的几何参数，这是古典和现代代数几何中的基本研究对象。PI期望退化技术可以成功地用于解决这些问题；事实上，现代工具(如半稳态约化和形变理论)使今天的退化方法比过去任何时候都更加强大。关于代数簇上曲线的几何，有许多悬而未决的问题，这里所描述的就是一个例子。在其中一个方面取得突破，往往会给其他人带来启发。这个项目就是这一事实的一个例子，因为这里使用的方法包含了PI用来在枚举代数几何的背景下证明结果的想法。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。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。

Caporaso 9801465 The goal of this project is to solve the following classical conjecture in algebraic geometry: the linear system of plane curves of fixed degree having a given number of singularities at assigned general points in the plane has dimension equal to the expected value. Such an expected value can be computed naively, and the above problem has been an object of study since the beginning of this century. Work that addresses this question would lead to a better understanding of the geometry of moduli spaces parameterizing plane curves, which are basic objects of study both in classical and modern algebraic geometry. The PI expects that degeneration techniques can be successfully used to approach these issues; in fact modern tools (such as semistable reduction and deformation theory) make degeneration methods much more powerful today than they were ever in the past. There are many open problems regarding the geometry of curves on algebraic varieties, of which the one described here is an example. It often happens that breakthroughs on one of them shed light on others. This project is an example of this fact, as the approach to be used here contains ideas that were used by the PI to prove results in the context of enumerative algebraic geometry. 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 its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.