Moduli of algebraic curves


In algebraic geometry, a moduli space of curves is a geometric space 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.

Moduli stacks of stable curves

The moduli stack classifies families of smooth projective curves, together with their isomorphisms. When, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves. 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. Both moduli stacks carry universal families of curves.
Both stacks above have dimension ; hence a stable nodal curve can be completely specified by choosing the values of parameters, when. 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. Hence the dimension of 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 has dimension 0.

Construction and irreducibility

It is a non-trivial theorem, proved by Pierre Deligne and David Mumford, that the moduli stack is irreducible, meaning it cannot be expressed as the union of two proper substacks. They prove this by analyzing the locus of stable curves in the Hilbert scheme
of tri-canonically embedded curves which have Hilbert polynomial . Then, the stack
is a construction of the moduli space. Using Deformation Theorysection 1, Deligne and Mumford show this stack is smooth and use the stack
of isomorphisms between stable curves to show that has finite stabilizers, hence it is a Deligne–Mumford stack. Moreover, they find a stratification of assection 3
where
and analyze the components of . If there existed multiple components of, none of them would be complete. Also, any component of must contain non-singular curves. Consequently, the singular locus is connected, hence it is contained in a single component of. Furthermore, because every component intersects, all components must be contained in a single component, hence the coarse space is irreducible. From the general theory of algebraic stacks, this implies the stack quotient is irreducible.

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 ; 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

Determing the geometry of the moduli space of genus curves can be established using deformation Theory. The number of moduli for a genus curve, e.g., is given by the cohomology group
With Serre duality this cohomology group is isomorphic to
for the dualizing sheaf. But, using Riemann-Roch shows the degree of the canonical bundle is, so the degree of is, hence there are no global sections, meaning
showing there are no deformations of genus curves. This proves is just a single point, and the only genus curves is given by. The only technical difficulty is the automorphism group of is the algebraic group, which rigidifies once three points on are fixed, so most authors take to mean.

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-invariant
where. Topologically, is just the affine line, but it can be compactified to a stack with underlying topological space by adding a stable curve at infinity. This is an elliptic curve with a single cusp. The construction of the general case over 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 would have stabilizer group at the point 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, 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, the parameter space for such curves is given by
where corresponds to the locus.

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 are parameters for sections of. Then, the locus of sections which contain no triple root contains every curve represented by a point.

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, 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 morphism
and it was long expected this would be true in all genera. In fact, Severi was proved this to be true for genera up to. It turns out though for genus 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 for. In fact, for,
and hence 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.

Stratification of boundary of \mathcal{M}_g

The moduli space has a natural stratification on the boundary whose points represent singular genus curves. 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 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 genus g curves with n marked points are denoted , and have dimension.
A case of particular interest is the moduli stack 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.

Boundary geometry

An important property of the compactified moduli spaces is that their boundary can be described in terms of moduli spaces for genera. 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 is isomorphic to the stack quotient of a product 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 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 are called "rational tail" and their moduli space is denoted. Stable curves whose dual graph is a tree are called "compact type" and their moduli space is denoted.

Classic references

Books on moduli of curves

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