Editing Moduli space (section)

{{short description|Geometric space whose points represent algebro-geometric objects of some fixed kind}}
In [[mathematics]], in particular [[algebraic geometry]], a '''moduli space'''  is a geometric space (usually a [[scheme (mathematics)|scheme]] or an [[algebraic stack]]) whose points represent algebro-geometric objects of some fixed kind, or [[isomorphism class]]es of such objects.  Such spaces frequently arise as solutions to classification problems:  If one can show that a collection of interesting objects (e.g., the smooth [[algebraic curve]]s of a fixed [[genus (topology)|genus]]) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space.  In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is [[formal moduli]]. [[Bernhard Riemann]] first used the term "moduli" in 1857.<ref>{{cite web |last1=Chan |first1=Melody |title=Moduli Spaces of Curves: Classical and Tropical |url=https://www.ams.org/journals/notices/202110/rnoti-p1700.pdf |website=AMS}}</ref>