The Geometry of Curves in Projective Space via Degeneration and Deformation

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Systems of polynomial equations are ubiquitous in mathematics, and related fields such as physics, cryptography, and other sciences. When such a system describes a one-dimensional object, the corresponding geometric object is known as an algebraic curve. Broadly speaking, this project studies the geometry of algebraic curves, both individually and as they vary in families. One specific question this project investigates is the concept of interpolation: When can a certain type of curve be passed through a general collection of points? In other words, one can draw a line through any two points in the plane and draw a circle through any three points in the plane, unless those three points lie on a line. More generally, the PI will consider what happens for other types of curves in higher-dimensional spaces, assuming again that the points do not lie in any special configurations. This project also includes training for undergraduate students. The project considers various questions for the investigation of the structure of the equations of general curves. For example, in what degrees are they generated? What is the Betti table of a general divisor on such a curve? The PI will also investigate certain natural vector bundles on curves that control their deformation theory --- the restricted tangent bundle and the normal bundle --- and study how they break down into stable vector bundles, which are the atomic building blocks of all vector bundles. Finally, the PI will study the geometry of moduli spaces of curves, focusing on their integral intersection theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.