Geometry of Linear Systems on Curves

这一建议涉及抽象和射影空间中的代数曲线的几何。在过去的几十年里，我们已经了解到抽象曲线如何嵌入到射影空间中；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到射影空间中。但我们仍然对嵌入曲线的几何形状知之甚少：什么样的方程定义了它们，以及发生了什么次数和类属。这些问题很耐人寻味，并有许多潜在的应用。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的几何产生影响。300多年来，代数曲线一直是研究的对象。在某种意义上，当数学家们第一次从研究一元多项式方程--只有有限数量的解--转移到两个变量的方程时，这个主题就开始了，其解形成了一条曲线。对这些几何对象的研究一直是我们理解多项式方程代数的主要来源。它导致了数学上的许多进步，包括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.