Geometry of Linear Systems on Curves

Harvard University, Cambridge MA

Investigators

Abstract

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.