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.
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The subset of the unmarked moduli space of Riemann surfaces with hyperbolic systole at least is compact for fixed genus . Use a Bers pants decomposition theorem bound, finitely many topological decomposition types and Dehn twists to place representatives in finitely many compact boxes in Fenchel–Nielsen coordinates.