Moduli spaces in algebraic geometry: construction and examples

Polynomial equations are among the simplest types of mathematical expressions. Examples include x^2+y^2 = 0 and x-y=0. At a basic level, these equations involve only addition and multiplication. Despite their simplicity, they are ubiquitous across the sciences, appearing in fields ranging from computer science and physics to chemistry and biology. When one tries to solve these equations, one often finds not just one answer, but an infinite number of them. To make sense of this, mathematicians look at the shape formed by all the solutions together. These shapes are called an algebraic variety. Major conjectures in modern Algebraic Geometry suggest that all algebraic varieties can be built, at least roughly, from just three basic types: Fano varieties, Calabi-Yau varieties, and varieties of general type. This project will describe how these three building blocks fit together. The project also includes research training opportunities for students. In more detail, the PI will (1) construct and investigate specific moduli spaces of fibered varieties, (2) explore the birational geometry of algebraic stacks, and (3) study both the local and global geometry of certain moduli spaces of varieties of general type and polarized Calabi-Yau varieties. Regarding (1), the PI will generalize existing frameworks for constructing moduli spaces of maps from families of curves to Deligne-Mumford stacks, and study specific examples of these moduli spaces. A key technique in this work involves the study of weighted blow-ups, a type of birational transformation for algebraic stacks that parallels classical blow-ups in the theory of projective varieties. Regarding (2), the PI plans to adapt and extend current methods in birational geometry to the more general setting of algebraic stacks. Finally, regarding (3), the PI will study the local and global geometric properties of moduli spaces of varieties of log-general type and Calabi-Yau varieties, contributing to a deeper understanding of their structure and classification. 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.