Linear Systems of Plane Curves With Assigned Singularities

本课题的目标是解决以下代数几何中的经典猜想：在平面上指定的一般点上具有给定奇异数的固定次平面曲线线性系统的维数等于期望值。这样的期望值可以被天真地计算出来，而上述问题自本世纪初以来一直是研究的对象。解决这个问题的工作将导致更好地理解模空间参数化平面曲线的几何，这是经典和现代代数几何的基本研究对象。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（如半稳定还原和变形理论）使退化方法比过去更加强大。关于代数变异体上的曲线几何，有许多尚未解决的问题，这里所描述的就是一个例子。通常情况下，其中一个领域的突破会给其他领域带来启发。这个项目就是这个事实的一个例子，因为这里使用的方法包含了PI在枚举代数几何的背景下用来证明结果的想法。这是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，但在过去的四分之一个世纪里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

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.