Moduli of algebraic curves

In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.
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").
Moduli stacks of stable curves.
The moduli stack formula_1 classifies families of smooth projective curves, together with their isomorphisms. When formula_2, 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. The resulting stack is denoted formula_3. Both moduli stacks carry universal families of curves.
Both stacks above have dimension formula_4; hence a stable nodal curve can be completely specified by choosing the values of formula_4 parameters, when formula_2. 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 formula_7 is equal to
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 formula_9 has dimension 0.
Construction and irreducibility.
It is a non-trivial theorem, proved by Pierre Deligne and David Mumford, that the moduli stack formula_10 is irreducible, meaning it cannot be expressed as the union of two proper substacks. They prove this by analyzing the locus formula_11 of stable curves in the Hilbert scheme
of tri-canonically embedded curves (from the embedding of the very ample formula_13 for every curve) which have Hilbert polynomial formula_14 (note: this can be computed using the Riemann–Roch theorem). Then, the stack
is a construction of the moduli space formula_10. Using Deformation Theorysection 1, Deligne and Mumford show this stack is smooth and use the stack
of isomorphisms between stable curves to show that formula_10 has finite stabilizers, hence it is a Deligne–Mumford stack (named after their paper). Moreover, they find a stratification of formula_11 assection 3
where
and analyze the components of formula_24 (as a GIT quotient). If there existed multiple components of formula_21, none of them would be complete. Also, any component of formula_11 must contain non-singular curves. Consequently, the singular locus formula_27 is connected, hence it is contained in a single component of formula_11. Furthermore, because every component intersects formula_27, all components must be contained in a single component, hence the coarse space formula_11 is irreducible. From the general theory of algebraic stacks, this implies the stack quotient formula_10 is irreducible.
Properness.
Properness, or compactness for orbifolds, follows from a theorem on stable reduction on curves. This can be found using a theorem of Grothendieck regarding the stable reduction of Abelian varieties, and showing its equivalence to the stable reduction of curves.section 5.2
Coarse moduli spaces.
One can also consider the 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 introduced. In fact, the idea of a moduli stack was introduced 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.
The coarse moduli spaces have the same dimension as the stacks when formula_2; however, in genus zero the coarse moduli space has dimension zero, and in genus one, it has dimension one.
Examples of low genus moduli spaces.
Genus 0.
Determining the geometry of the moduli space of genus formula_33 curves can be established using deformation Theory. The number of moduli for a genus formula_33 curve, e.g. formula_35, is given by the cohomology groupformula_36With Serre duality this cohomology group is isomorphic toformula_37for the dualizing sheaf formula_38. But, using Riemann-Roch shows the degree of the canonical bundle is formula_39, so the degree of formula_40 is formula_41, hence there are no global sections, meaningformula_42showing there are no deformations of genus formula_33 curves. This proves formula_7 is just a single point, and the only genus formula_33 curves is given by formula_35. The only technical difficulty is the automorphism group of formula_35 is the algebraic group formula_48, which rigidifies once three points on formula_35 are fixed, so most authors take formula_7 to mean formula_51.
Genus 1.
The genus 1 case is one of the first well-understood cases of moduli spaces, at least over the complex numbers, because isomorphism classes of elliptic curves are classified by the J-invariantformula_52where formula_53. Topologically, formula_54 is just the affine line, but it can be compactified to a stack with underlying topological space formula_55 by adding a stable curve at infinity. This is an elliptic curve with a single cusp. The construction of the general case over formula_56 was originally completed by Deligne and Rapoport.
Note 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 formula_9 would have stabilizer group at the point formula_58 given by the curve, since elliptic curves have an Abelian group structure. This adds unneeded technical complexity to this hypothetical moduli space. On the other hand, formula_59 is a smooth Deligne–Mumford stack.
Genus 2.
Affine parameter space.
In genus 2 it is a classical result that all such curves are hyperelliptic,pg 298 so the moduli space can be determined completely from the branch locus of the curve using the Riemann–Hurwitz formula. Since an arbitrary genus 2 curve is given by a polynomial of the form
for some uniquely defined formula_61, the parameter space for such curves is given by
where formula_63 corresponds to the locus formula_64.
Weighted projective space.
Using a weighted projective space and the Riemann–Hurwitz formula, a hyperelliptic curve can be described as a polynomial of the form
where formula_66 are parameters for sections of formula_67. Then, the locus of sections which contain no triple root contains every curve formula_68 represented by a point formula_69.
Genus 3.
This is the first moduli space of curves which has both a hyperelliptic locus and a non-hyperelliptic locus. 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
Birational geometry.
Unirationality conjecture.
In all of the previous cases, the moduli spaces can be found to be unirational, meaning there exists a dominant rational morphismformula_72and it was long expected this would be true in all genera. In fact, Severi was proved this to be true for genera up to formula_73. It turns out though for genus formula_74 all such moduli spaces are of general type, meaning they are not unirational. They accomplished this by studying the Kodaira dimension of the coarse moduli spaces
and found formula_76 for formula_74. In fact, for formula_78,
and hence formula_10 is of general type.
Geometric implication.
This is significant geometrically because it implies any linear system on a ruled variety cannot contain the universal curve formula_81.
Stratification of boundary of formula_10.
The moduli space formula_3 has a natural stratification on the boundary formula_84 whose points represent singular genus formula_85 curves. It decomposes into strata
where
The curves lying above these loci correspond to
Stratification of formula_94.
For the genus formula_95 case, there is a stratification given by
Further analysis of these strata can be used to give the generators of the Chow ring formula_97 proposition 9.1.
Moduli of marked curves.
One can also enrich the problem by considering the moduli stack of genus g nodal curves with n marked points, pairwise distinct and distinct from the nodes. 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 formula_98 (or formula_99), and have dimension formula_100.
A case of particular interest is the moduli stack formula_101 of genus 1 curves with one marked point. This is the stack of elliptic curves. 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").
Boundary geometry.
An important property of the compactified moduli spaces formula_99 is that their boundary can be described in terms of moduli spaces formula_103 for genera formula_104. 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. Here the vertices of the graph correspond to irreducible components of the nodal curve, the labelling of a vertex is the arithmetic genus of the corresponding component, edges correspond to nodes of the curve and the half-edges correspond to the markings. The closure of the locus of curves with a given dual graph in formula_99 is isomorphic to the stack quotient of a product formula_106 of compactified moduli spaces of curves by a finite group. In the product the factor corresponding to a vertex "v" has genus gv taken from the labelling and number of markings formula_107 equal to the number of outgoing edges and half-edges at "v". The total genus "g" is the sum of the gv plus the number of closed cycles in the graph.
Stable curves whose dual graph contains a vertex labelled by formula_108 (hence all other vertices have formula_109 and the graph is a tree) are called "rational tail" and their moduli space is denoted formula_110. Stable curves whose dual graph is a tree are called "compact type" (because the Jacobian is compact) and their moduli space is denoted formula_111.