Commutative Algebra methods for Hilbert schemes and beyond

Polynomial equations are ubiquitous in science, describing important physical principles and serving as mathematical models for complex natural phenomena. Algebraic geometry studies geometric structures arising from solutions to systems of polynomial equations. To gain a better understanding of these structures, it is useful to study how they change when the corresponding equations are slightly perturbed. This is achieved by studying a “parameter space” for these structures. The overarching goal of this project is to use techniques from commutative algebra to tackle longstanding questions related to the Hilbert scheme, a parameter space for polynomials with fixed properties. The project’s broader impacts include developing new packages for the open-source computer algebra system Macaulay2, organizing local seminars, and organizing mathematical conferences. The investigator will focus on three areas of commutative algebra and algebraic geometry: 1) Singularities of the Hilbert scheme of points on a threefold: The main goal is to understand the singularities of the Hilbert scheme of points on a smooth threefold. In particular, the investigator will focus on determining the smooth points and explaining some of the patterns appearing in the structure of the singularities. 2) Exploring multigraded Hilbert schemes and other moduli spaces: The investigator will study the space of branch varieties, a close analogue of the Hilbert scheme, and focus on studying the projectivity of this moduli space. 3) Varieties in weighted projective spaces: The investigator will focus on developing a set of tools to extend classical theorems in projective space, such as Macaulay’s theorem on the existence of Hilbert functions and the del Pezzo-Bertini classification of varieties of minimal degree, to weighted projective spaces. 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.