Editing Moduli space (section)

===Moduli of curves===
{{details|Moduli of algebraic curves}}

The moduli stack <math>\mathcal{M}_{g}</math> classifies families of smooth projective curves of genus ''g'', together with their isomorphisms. When ''g'' > 1, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves (together with their isomorphisms). A curve is stable if it has only a finite group of automorphisms. The resulting stack is denoted <math>\overline{\mathcal{M}}_{g}</math>. Both moduli stacks carry universal families of curves. One can also define coarse moduli spaces representing isomorphism classes of smooth or stable curves. These coarse moduli spaces were actually studied before the notion of moduli stack was invented. In fact, the idea of a moduli stack was invented by Deligne and Mumford in an attempt to prove the projectivity of the coarse moduli spaces. In recent years, it has become apparent that the stack of curves is actually the more fundamental object.

Both stacks above have dimension 3''g''−3; hence a stable nodal curve can be completely specified by choosing the values of 3''g''−3 parameters, when ''g'' > 1. In lower genus, one must account for the presence of smooth families of automorphisms, by subtracting their number. There is exactly one complex curve of genus zero, the Riemann sphere, and its group of isomorphisms is PGL(2). Hence, the dimension of <math>\mathcal{M}_0</math> is

: dim(space of genus zero curves) − dim(group of automorphisms)  = 0 − dim(PGL(2)) = −3.

Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional group of automorphisms. Hence, the stack <math>\mathcal{M}_1</math> has dimension 0. The coarse moduli spaces have dimension 3''g''−3 as the stacks when ''g'' > 1 because the curves with genus g > 1 have only a finite group as its automorphism i.e. dim(a group of automorphisms) = 0. Eventually, in genus zero, the coarse moduli space has dimension zero, and in genus one, it has dimension one.

One can also enrich the problem by considering the moduli stack of genus ''g'' nodal curves with ''n'' marked points. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite. The resulting moduli stacks of smooth (or stable) genus ''g'' curves with ''n''-marked points are denoted <math>\mathcal{M}_{g,n}</math> (or <math>\overline{\mathcal{M}}_{g,n}</math>), and have dimension 3''g''&nbsp;−&nbsp;3&nbsp;+&nbsp;''n''.

A case of particular interest is the moduli stack <math>\overline{\mathcal{M}}_{1,1}</math> of genus 1 curves with one marked point.  This is the stack of [[elliptic curve]]s, and is the natural home of the much studied [[modular form]]s, which are meromorphic sections of bundles on this stack.