Geometry of Linear Systems on Curves

这个建议涉及几何代数曲线，无论是在抽象和射影空间。在过去的几十年里，我们已经学到了很多关于抽象曲线如何嵌入射影空间的知识;特别是，我们知道给定亏格的一般曲线何时可以作为给定次数的曲线嵌入射影空间。但是我们仍然不太了解嵌入曲线的几何形状：什么样的方程定义它们，以及发生了什么程度和一般。这些问题很有趣，并且有许多潜在的应用。例如，理解定义嵌入曲线的多项式--特别是解决最大秩猜想--将反过来对抽象曲线的模空间几何产生影响。代数曲线已经成为研究对象300多年了。从某种意义上说，这一主题始于数学家们第一次从研究一个变量的多项式方程（只有有限数量的解）转向研究两个变量的方程（其解形成一条曲线）。对这些几何对象的研究是我们理解多项式方程代数的主要来源。它导致了数学的许多进步，无论是在代数几何和许多其他数学领域，如拓扑学，数论，复分析和数学物理。

This proposal concerns the geometry of algebraic curves, both in the abstract and in projective space. Over the last few decades, we have learned a great deal about how abstract curves may be embedded in projective space; in particular, we know when a general curve of given genus can be embedded in projective space as a curve of given degree. But we still don't know as much about the geometry of the embedded curves: what sort of equations define them, and what degrees and genera occur. These questions are intriguing, and have many potential applications. For example, understanding the polynomials that define the embedded curves -- in particular, resolving the Maximal Rank Conjecture -- will have consequences in turn for the geometry of moduli spaces of abstract curves.Algebraic curves have been the object of study for more than 300 years. In some sense, the subject started when mathematicians first moved from studying polynomial equations in one variable -- which have just a finite number of solutions -- to equations in two variables, whose solutions form a curve. The study of these geometric objects has been the main source of our understanding of the algebra of polynomial equations. It has led to many advances in mathematics, both in algebraic geometry and many other areas of mathematics, such as topology, number theory, complex analysis and mathematical physics.