Moduli of algebraic curves

In algebraic geometry, a moduli space of curves is a space whose points correspond to isomorphism classes of algebraic curves. The term "modulus" was introduced for this purpose by Bernhard Riemann, and means "parameter"; thus a "moduli space" means a space giving parameters that specify all of the curves of a given kind. With a moduli space, instead of studying one curve at a time, one studies all curves of a given kind as members of a single geometric family. The moduli space of curves (of a given kind) is a special case of the more general notion moduli space, which gives a parameter space for other kinds of objects (curves, surfaces, etc).

Different choices of conditions lead to different moduli spaces. For example, one may fix the genus, allow only smooth or also certain singular curves, or include marked points. Depending on the problem, the moduli object may be constructed as a scheme, an algebraic space, or more naturally as an algebraic stack. In many cases there is both a coarse moduli space, which records isomorphism classes of curves, and a finer stack that also keeps track of their automorphisms.

A well-studied example is the moduli of smooth projective curves of genus ${\displaystyle g}$. Over the field of complex numbers, these correspond to compact Riemann surfaces. Classically, the (coarse) moduli space of genus ${\displaystyle g=1}$ curves having a marked point (elliptic curve groups) is the (classical) modular curve. For ${\displaystyle g>1}$, the moduli stack of smooth curves is denoted ${\displaystyle {\mathcal {M}}_{g}}$, and its compactification by stable nodal curves is denoted ${\displaystyle {\overline {\mathcal {M}}}_{g}}$. These spaces and stacks play a central role in algebraic geometry, Teichmüller theory, and the theory of modular forms.