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").

## Contents

* 1 Moduli stacks of stable curves
* 1.1 Construction and irreducibility
* 1.2 Properness
* 1.3 Coarse moduli spaces
* 2 Examples of low genus moduli spaces
* 2.1 Genus 0
* 2.2 Genus 1
* 2.3 Genus 2
* 2.3.1 Affine parameter space
* 2.3.2 Weighted projective space
* 2.4 Genus 3
* 3 Birational geometry
* 3.1 Unirationality conjecture
* 3.1.1 Geometric implication
* 4 Stratification of boundary
* 4.1 Stratification for genus 2
* 5 Moduli of marked curves
* 6 Boundary geometry
* 7 See also
* 8 References
* 8.1 Classic references
* 8.2 Books on moduli of curves
* 8.3 Cohomology and intersection theory
* 9 External links

## Moduli stacks of stable curves

The moduli stack ℳg classifies families of smooth projective curves, 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 is complete, connected, has no singularities other than double points, and has only a finite group of automorphisms. The resulting stack is denoted ℳ‾g. Both moduli stacks carry universal families of curves.

Both stacks above have dimension 3g−3; hence a stable nodal curve can be completely specified by choosing the values of 3g−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 ℳ0 is equal to

dim⁡(space of genus 0 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 ℳ1 has dimension 0.

### Construction and irreducibility

It is a non-trivial theorem, proved by Pierre Deligne and David Mumford,[1] that the moduli stack ℳg is irreducible, meaning it cannot be expressed as the union of two proper substacks. They prove this by analyzing the locus Hg of stable curves in the Hilbert scheme Hilbℙ5g−5−1Pg(n) of tri-canonically embedded curves (from the embedding of the very ample ωC⊗3 for every curve) which have Hilbert polynomial Pg(n)=(6n−1)(g−1). Then, the stack [Hg/PGL(5g−6)] is a construction of the moduli space ℳg. Using deformation theory, Deligne and Mumford show this stack is smooth and use the stack of isomorphisms between stable curves IsomS(C,C′), to show that ℳg has finite stabilizers, hence it is a Deligne–Mumford stack. Moreover, they find a stratification of Hg as

Hgo∐Hg,1∐⋯∐Hg,n,

where Hgo is the subscheme of smooth stable curves and Hg,i is an irreducible component of S*=Hg∖Hgo. They analyze the components of ℳg0=Hg0/PGL(5g−6) (as a GIT quotient). If there existed multiple components of Hgo, none of them would be complete. Also, any component of Hg must contain non-singular curves. Consequently, the singular locus S* is connected, hence it is contained in a single component of Hg. Furthermore, because every component intersects S*, all components must be contained in a single component, hence the coarse space Hg is irreducible. From the general theory of algebraic stacks, this implies the stack quotient ℳg is irreducible.

### Properness

Properness, or compactness for orbifolds, follows from a theorem on stable reduction on curves.[1] 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.[1]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 g>1; 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

curves can be established using deformation Theory. The number of moduli for a genus

curve, e.g.

, is given by the cohomology group

> H1(C,TC)

With Serre duality this cohomology group is isomorphic to

> H1(C,TC)≅H0(C,ωC⊗TC∨)≅H0(C,ωC⊗2)

for the dualizing sheaf

. But, using Riemann–Roch shows the degree of the canonical bundle is

, so the degree of

ωC⊗2

, hence there are no global sections, meaning

> H0(C,ωC⊗2)=0

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

PGL(2,ℂ)

, which rigidifies once three points[2] on

are fixed, so most authors take

to mean

ℳ0,3

### Genus 1

Main article: Moduli stack of elliptic curves

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

> j:ℳ1,1|ℂ->𝔸ℂ1

where

ℳ1,1|ℂ=ℳ1,1×Spec(ℤ)Spec(ℂ)

. Topologically,

ℳ1,1|ℂ

is just the affine line, but it can be compactified to a stack with underlying topological space

ℙℂ1

by adding a stable curve at infinity. This is an elliptic curve with a single cusp. The construction of the general case over

Spec(ℤ)

was originally completed by Deligne and Rapoport.[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 ℳ1 would have stabilizer group at the point [C]∈ℳ1 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, ℳ1,1 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,[4]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

y2−x(x−1)(x−a)(x−b)(x−c)

for some uniquely defined a,b,c∈𝔸1, the parameter space for such curves is given by

𝔸3∖(Δa,b∪Δa,c∪Δb,c),

where Δi,j corresponds to the locus i≠j.[5]

#### 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[6]

z2=ax6+bx5y+cx4y2+dx3y3+ex2y4+fxy5+gy6,

where a,…,f are parameters for sections of Γ(ℙ(3,1),𝒪(g)). Then, the locus of sections which contain no triple root contains every curve C represented by a point [C]∈ℳ2.

### Genus 3

This is the first moduli space of curves which has both a hyperelliptic locus and a non-hyperelliptic locus.[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

Hilbℙ28t−4≅ℙ(64)−1.

Then, the moduli space is stratified by the substacks

ℳ3=[H2/PGL(3))]∐ℳ3hyp.

## 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

> ℙn->ℳg

and it was long expected this would be true in all genera. In fact, Severi had proved this to be true for genera up to

.[9] Although, it turns out that for genus

g≥23

[10][11][12] 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

κg=Kod(ℳ‾g),

and found κg>0 for g≥23. In fact, for g>23,

κg=3g−3=dim⁡(ℳg),

and hence ℳg 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 𝒞g.[13]

## Stratification of boundary

The moduli space ℳ‾g has a natural stratification on the boundary ∂ℳ‾g whose points represent singular genus g curves.[14] It decomposes into strata

∂ℳ‾g=∐0≤h≤(g/2)Δh*,

where

* Δh*≅ℳ‾h×ℳ‾g−h for 1≤h<g/2.
* Δ0*≅ℳ‾g−1,2/(ℤ/2) where the action permutes the two marked points.
* Δg/2≅(ℳ‾g/2×ℳ‾g/2)/(ℤ/2) whenever g is even.

The curves lying above these loci correspond to

* A pair of curves C,C′ connected at a double point.
* The normalization of a genus g curve at a single double point singularity.
* A pair of curves of the same genus connected at a double point up to permutation.

### Stratification for genus 2

For the genus 2 case, there is a stratification given by

∂ℳ‾2=Δ0*∐Δ1*=ℳ‾1,2/(ℤ/2)∐(ℳ‾1×ℳ‾1)/(ℤ/2).

Further analysis of these strata can be used to give the generators of the Chow ring A*(ℳ‾2)[14] 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 ℳg,n (or ℳ‾g,n), and have dimension 3g−3+n.

A case of particular interest is the moduli stack ℳ‾1,1 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 ℳ‾g,n is that their boundary can be described in terms of moduli spaces ℳ‾g′,n′ for genera g′<g. 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 ℳ‾g,n is isomorphic to the stack quotient of a product ∏vℳ‾gv,nv 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 nv 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 gv=g (hence all other vertices have gv=0 and the graph is a tree) are called "rational tail" and their moduli space is denoted ℳg,nr.t.. Stable curves whose dual graph is a tree are called "compact type" (because the Jacobian is compact) and their moduli space is denoted ℳg,nc..[2]

## See also

* Moduli of marked curves
* Witten conjecture
* Tautological ring
* Grothendieck–Riemann–Roch theorem

## References

1. ↑ 1.0 1.1 1.2 Deligne, Pierre; Mumford, David (1969). "The irreducibility of the space of curves of given genus" (in en). Publications Mathématiques de l'IHÉS 36: 75–109. doi:10.1007/BF02684599. http://www.numdam.org/item/?id=PMIHES_1969__36__75_0.
2. ↑ 2.0 2.1 Faber, Carel; Pandharipande, Rahul (2011). "Tautological and non-tautological cohomology of the moduli space of curves". arXiv:1101.5489 [math.AG].
3. ↑ Deligne, P.; Rapoport, M. (1973), Les schémas de modules de courbes elliptiques, Lecture Notes in Mathematics, 349, Springer Berlin Heidelberg, pp. 143–316, doi:10.1007/bfb0066716, ISBN 978-3-540-06558-6 , URL: http://publications.ias.edu/node/367
4. ↑ Hartshorne, Robin (29 June 2013). Algebraic geometry. New York. ISBN 978-1-4757-3849-0. OCLC 861706007.
5. ↑ Igusa, Jun-Ichi (1960). "Arithmetic Variety of Moduli for Genus Two". Annals of Mathematics 72 (3): 612–649. doi:10.2307/1970233. ISSN 0003-486X.
6. ↑ Larson, Eric (2019-04-17). "The integral Chow ring of M‾2". arXiv:1904.08081 [math.AG].
7. ↑ Girard, Martine; Kohel, David R. (2006), Hess, Florian; Pauli, Sebastian; Pohst, Michael, eds., "Classification of Genus 3 Curves in Special Strata of the Moduli Space", Algorithmic Number Theory (Berlin, Heidelberg: Springer Berlin Heidelberg) 4076: pp. 346–360, doi:10.1007/11792086_25, ISBN 978-3-540-36075-9, Bibcode: 2006math......3555G
8. ↑ Penev, Nikola; Vakil, Ravi (2015). "The Chow ring of the moduli space of curves of genus six". Algebraic Geometry 2 (1): 123–136. doi:10.14231/ag-2015-006. ISSN 2214-2584.
9. ↑ Severi, Francesco, 1879-1961. (1915). Sulla classificazione delle curve algebriche e sul teorema d'esistenza di Riemann. Tipografia della R. Accademia dei Lincei. OCLC 881814709.
10. ↑ Eisenbud, David; Harris, Joe (1987). "The Kodaira dimension of the moduli space of curves of genus ?23". Inventiones Mathematicae 90 (2): 359–387. doi:10.1007/bf01388710. ISSN 0020-9910. Bibcode: 1987InMat..90..359E.
11. ↑ Harris, Joe; Mumford, David (1982), "On the Kodaira Dimension of the Moduli Space of Curves", Selected Papers (New York, NY: Springer New York): pp. 171–234, doi:10.1007/978-1-4757-4265-7_8, ISBN 978-1-4419-1936-6, https://dash.harvard.edu/bitstream/1/3613574/3/Mumford_OnKodairaDim.pdf
12. ↑ Harris, Joe; Mumford, David (1982), "On the Kodaira Dimension of the Moduli Space of Curves", Selected Papers (New York, NY: Springer New York): pp. 171–234, doi:10.1007/978-1-4757-4265-7_8, ISBN 978-1-4419-1936-6, http://nrs.harvard.edu/urn-3:HUL.InstRepos:3613574
13. ↑ Farkas, Gavril (2009). "The global geometry of the moduli space of curves". Algebraic Geometry. Proceedings of Symposia in Pure Mathematics. 80. pp. 125–147. doi:10.1090/pspum/080.1/2483934. ISBN 9780821847022.
14. ↑ 14.0 14.1 Arithmetic and geometry: papers dedicated to I.R. Shafarevich on the occasion of his sixtieth birthday. Shafarevich, Igor Rostislavovich, 1923-2017, Artin, Michael, Tate, John Torrence, 1925-2019. Boston: Birkhäuser. 1983. ISBN 978-1-4757-9286-7. OCLC 681426064. https://www.dam.brown.edu/people/mumford/alg_geom/papers/1983b--EnumGeomModuli-NC.pdf.

### Classic references

* Grothendieck, Alexander (1960–1961). "Techniques de construction en géométrie analytique. I. Description axiomatique de l'espace de Teichmüller et de ses variantes". Séminaire Henri Cartan (Paris) 13 (1). Exposés No. 7 and 8. http://archive.numdam.org/article/SHC_1960-1961__13_1_A4_0.pdf.

* Mumford, David; Fogarty, John; Kirwan, Frances Clare (1994). Geometric invariant theory (3rd enl. ed.). Berlin: Springer-Verlag. ISBN 3-540-56963-4. OCLC 29184987.
* "The irreducibility of the space of curves of given genus". Publications Mathématiques de l'IHÉS 36: 75–109. 1969. doi:10.1007/bf02684599. http://archive.numdam.org/article/PMIHES_1969__36__75_0.pdf.

### Books on moduli of curves

* Harris, Joe; Morrison, Ian (1998). Moduli of Curves. Springer Verlag. ISBN 978-0-387-98429-2.

* Katz, Nicholas M (1985). Arithmetic Moduli of Elliptic Curves. Princeton University Press. ISBN 978-0-691-08352-0.
* Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip A. (2011). Geometry of Algebraic Curves II. Grundlehren der mathematischen Wissenschaften. 268. doi:10.1007/978-3-540-69392-5. ISBN 978-3-540-42688-2.

### Cohomology and intersection theory

* Zvonkine, Dimitri (2012). "An introduction to moduli spaces of curves and their intersection theory". in Papadopoulos, Athanase. Handbook of Teichmüller Theory, Volume III. IRMA Lectures in Mathematics and Theoretical Physics. 17. Zürich, Switzerland: European Mathematical Society Publishing House. pp. 667–716. doi:10.4171/103-1/12. ISBN 978-3-03719-103-3. https://www-fourier.ujf-grenoble.fr/sites/ifmaquette.ujf-grenoble.fr/files/ete2011-zvonkine.pdf.
* Faber, Carel; Pandharipande, Rahul (2013). Farkas, Gavril; Morrison, Ian. eds. Tautological and non-tautological cohomology of the moduli space of curves. Advanced Lectures in Mathematics (ALM). 24. Somerville, MA: International Press. pp. 293–330. ISBN 9781571462572. https://people.math.ethz.ch/~rahul/tnt.pdf.

## External links

* "Topology and geometry of the moduli space of curves". https://aimath.org/WWN/modspacecurves/.
* "Moduli of Stable Maps, Gromov-Witten Invariants, and Quantum Cohomology"

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Topics in algebraic curves

Rational curves|

* Five points determine a conic
* Projective line
* Rational normal curve
* Riemann sphere
* Twisted cubic

Elliptic curves|

| Analytic theory|

* Elliptic function
* Elliptic integral
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* Modular form

|

Arithmetic theory|

* Counting points on elliptic curves
* Division polynomials
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* Mazur's torsion theorem
* Modular elliptic curve
* Modularity theorem
* Mordell–Weil theorem
* Nagell–Lutz theorem
* Supersingular elliptic curve
* Schoof's algorithm
* Schoof–Elkies–Atkin algorithm

Applications|

* Elliptic curve cryptography
* Elliptic curve primality

Higher genus|

* De Franchis theorem
* Faltings's theorem
* Hurwitz's automorphisms theorem
* Hurwitz surface
* Hyperelliptic curve

Plane curves|

* AF+BG theorem
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Riemann surfaces|

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Constructions|

* Dual curve
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Structure of curves|

| Divisors on curves|

* Abel–Jacobi map
* Brill–Noether theory
* Clifford's theorem on special divisors
* Gonality of an algebraic curve
* Jacobian variety
* Riemann–Roch theorem
* Weierstrass point
* Weil reciprocity law

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Moduli|

* ELSV formula
* Gromov–Witten invariant
* Hodge bundle
* Moduli of algebraic curves
* Stable curve

Morphisms|

* Hasse–Witt matrix
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* Prym variety
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Singularities|

* Acnode
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Vector bundles|

* Birkhoff–Grothendieck theorem
* Stable vector bundle
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