The basic algebra-geometry dictionary for finitely generated k-algebras is one of the triumphs of 19th and early 20th century mathematics. However, classes of related rings, such as their k-subalgebras, lack clean general properties or organizing principles, even when they arise naturally in problems of smooth projective geometry. “Stabilization” in smooth topology and symplectic geometry, achieved by products with Euclidean space, substantially simplifies many
problems. We discuss an analog in the more rigid setting of algebraic and arithmetic geometry, which, among other things (e.g., applications to counting rational points), gives some structure to the study of k-subalgebras. We focus on the case of the moduli space of stable rational n-pointed curves to illustrate.
- Geometry and Analysis Seminar