Editing Moduli space (section)

==Further examples==

===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.

===Moduli of varieties===
In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higher-dimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties, such as the [[Siegel modular variety]]. This is the problem underlying [[Siegel modular form]] theory. See also [[Shimura variety]].

Using techniques arising out of the minimal model program, moduli spaces of varieties of general type were constructed by [[János Kollár]] and [[Nicholas Shepherd-Barron]], now known as KSB moduli spaces.<ref>J. Kollar. Moduli of varieties of general type, Handbook of moduli. Vol. II, 2013, pp. 131–157.</ref>

Using techniques arising out of differential geometry and birational geometry simultaneously, the construction of moduli spaces of [[Fano varieties]] has been achieved by restricting to a special class of [[K-stability of Fano varieties|K-stable]] varieties. In this setting important results about boundedness of Fano varieties proven by [[Caucher Birkar]] are used, for which he was awarded the 2018 [[Fields Medal|Fields medal]]. 

The construction of moduli spaces of Calabi-Yau varieties is an important open problem, and only special cases such as moduli spaces of [[K3 surface|K3 surfaces]] or [[Abelian varieties]] are understood.<ref>Huybrechts, D., 2016. ''Lectures on K3 surfaces'' (Vol. 158). Cambridge University Press.</ref>

===Moduli of vector bundles===
Another important moduli problem is to understand the geometry of (various substacks of) the moduli stack Vect<sub>''n''</sub>(''X'') of rank ''n'' [[vector bundle]]s on a fixed [[algebraic variety]] ''X''.<ref>{{Cite web|title=Algebraic Stacks and Moduli of Vector Bundles|url=https://impa.br/wp-content/uploads/2017/04/PM_36.pdf}}</ref> This stack has been most studied when ''X'' is one-dimensional, and especially when n equals one.  In this case, the coarse moduli space is the [[Picard scheme]], which like the moduli space of curves, was studied before stacks were invented. When the bundles have rank 1 and degree zero, the study of coarse moduli space is the study of the [[Jacobian variety]].

In applications to [[physics]], the number of moduli of vector bundles and the closely related problem of the number of moduli of [[Fiber bundle|principal G-bundles]] has been found to be significant in [[gauge theory]].{{citation needed|date=June 2013}}

=== Volume of the moduli space ===
Simple geodesics and Weil-Petersson [https://www.math.stonybrook.edu/~mlyubich/Archive/Geometry/Teichmuller%20Space/Mirz3.pdf volumes of moduli spaces] of bordered Riemann surfaces.