Mumford's compactness theorem

Mumford's compactness theorem is a result in algebraic geometry that pertains to the study of families of algebraic curves. Specifically, it provides conditions under which a certain space of algebraic curves can be compactified. The theorem states that the moduli space of stable curves of a given genus \( g \) (the space that parameterizes all algebraic curves of that genus, up to certain equivalences) is compact.