The most basic problem is that of moduli of smooth complete curves of a fixed genus.
Over the field of complex numbers these correspond precisely to compact Riemann surfaces of the given genus, for which Bernhard Riemann proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure depends").
classifies families of smooth projective curves, together with their isomorphisms.
, 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 is complete, connected, has no singularities other than double points, and has only a finite group of automorphisms.
Both moduli stacks carry universal families of curves.
; hence a stable nodal curve can be completely specified by choosing the values of
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).
It is a non-trivial theorem, proved by Pierre Deligne and David Mumford,[1] that the moduli stack
Properness, or compactness for orbifolds, follows from a theorem on stable reduction on curves.
[1]section 5.2 One can also consider the coarse moduli spaces representing isomorphism classes of smooth or stable curves.
In recent years, it has become apparent that the stack of curves is actually the more fundamental object.
is just the affine line, but it can be compactified to a stack with underlying topological space
[3] Note that most authors consider the case of genus one curves with one marked point as the origin of the group since otherwise the stabilizer group in a hypothetical moduli space
This adds unneeded technical complexity to this hypothetical moduli space.
Since an arbitrary genus 2 curve is given by a polynomial of the form for some uniquely defined
[5] Using a weighted projective space and the Riemann–Hurwitz formula, a hyperelliptic curve can be described as a polynomial of the form[6] where
[7][8] The non-hyperelliptic curves are all given by plane curves of degree 4 (using the genus degree formula), which are parameterized by the smooth locus in the Hilbert scheme of hypersurfaces Then, the moduli space is stratified by the substacks
In all of the previous cases, the moduli spaces can be found to be unirational, meaning there exists a dominant rational morphism
They accomplished this by studying the Kodaira dimension of the coarse moduli spaces and found
This is significant geometrically because it implies any linear system on a ruled variety cannot contain the universal curve
[14] It decomposes into strata where The curves lying above these loci correspond to For the genus
case, there is a stratification given by Further analysis of these strata can be used to give the generators of the Chow ring
The resulting moduli stacks of smooth (or stable) genus g curves with n marked points are denoted
Level 1 modular forms are sections of line bundles on this stack, and level N modular forms are sections of line bundles on the stack of elliptic curves with level N structure (roughly a marking of the points of order N).
Given a marked, stable, nodal curve one can associate its dual graph, a graph with vertices labelled by nonnegative integers and allowed to have loops, multiple edges and also numbered half-edges.
In the product the factor corresponding to a vertex v has genus gv taken from the labelling and number of markings
Stable curves whose dual graph contains a vertex labelled by
and the graph is a tree) are called "rational tail" and their moduli space is denoted