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((more citations needed|date=August 2019))
In [[algebraic geometry]], a '''moduli space of''' ('''algebraic''') '''curves'''  is a geometric space (typically a [[scheme (mathematics)|scheme]] or an [[algebraic stack]]) whose points represent isomorphism classes of [[algebraic curve]]s.  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  [[Moduli space#Fine Moduli Spaces|fine]] and [[Moduli space#Coarse Moduli Spaces|coarse moduli spaces]] for the same moduli problem.

The most basic problem is that of moduli of [[smooth morphism|smooth]] [[complete variety|complete]] curves of a fixed [[Genus (mathematics)|genus]]. Over the [[field (mathematics)|field]] of [[complex numbers]] these correspond precisely to [[compact Riemann surface]]s 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 <math>\mathcal{M}_{g}</math> classifies families of smooth projective curves, together with their isomorphisms.   When <math>g > 1</math>, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves (together with their isomorphisms).  A curve is [[stable curve|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 <math>\overline{\mathcal{M))_{g}</math>.  Both moduli stacks carry universal families of curves.

Both stacks above have dimension <math>3g-3</math>; hence a stable nodal curve can be completely specified by choosing the values of <math>3g-3</math> parameters, when <math>g > 1</math>.  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 equal to

:<math>\begin{align}\dim(\text{space of genus 0 curves}) - \dim(\text{group of automorphisms})  &= 0 - \dim(\mathrm{PGL}(2))\\ 
&= -3 .\end{align}</math>

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.

=== Construction and irreducibility ===
It is a non-trivial theorem, proved by [[Pierre Deligne]] and [[David Mumford]],<ref name=":0">((Cite journal|last1=Deligne|first1=Pierre|author1-link=Pierre Deligne|last2=Mumford|first2=David|author2-link=David Mumford|date=1969|title=The irreducibility of the space of curves of given genus|url=http://www.numdam.org/item/?id=PMIHES_1969__36__75_0|journal=[[Publications Mathématiques de l'IHÉS]]|language=en|volume=36|pages=75–109|doi=10.1007/BF02684599|s2cid=16482150))</ref> that the moduli stack <math>\mathcal{M}_g</math> is irreducible, meaning it cannot be expressed as the union of two proper substacks. They prove this by analyzing the locus <math>H_g</math> of [[stable curve]]s in the [[Hilbert scheme]] <math>\mathrm{Hilb}_{\mathbb{P}^{5g - 5 -1))^{P_g(n)}</math> of tri-canonically embedded curves (from the embedding of the very ample <math>\omega_C^{\otimes 3}</math> for every curve) which have [[Hilbert polynomial]] <math>P_g(n) = (6n-1)(g-1)</math>. Then, the stack <math>[H_g / \mathrm{PGL}(5g-6)]</math> is a construction of the moduli space <math>\mathcal{M}_g</math>. Using [[Deformation (mathematics)|deformation theory]], Deligne and Mumford show this stack is smooth and use the stack of isomorphisms between stable curves <math>\mathrm{Isom}_S(C,C')</math>, to show that <math>\mathcal{M}_g</math> has finite stabilizers, hence it is a [[Deligne–Mumford stack]]. Moreover, they find a stratification of <math>H_g</math> as 

:<math>H_g^o \coprod H_{g,1} \coprod \cdots \coprod H_{g,n}</math>,

where <math>H_g^o</math> is the subscheme of smooth stable curves and <math>H_{g,i}</math> is an irreducible component of <math>S^* = H_g \setminus H_g^o</math>. They analyze the components of <math>\mathcal{M}_g^0 = H_g^0/\mathrm{PGL}(5g-6)</math> (as a [[GIT quotient]]). If there existed multiple components of <math>H_g^o</math>, none of them would be complete. Also, any component of <math>H_g</math> must contain non-singular curves. Consequently, the singular locus <math>S^*</math> is connected, hence it is contained in a single component of <math>H_g</math>. Furthermore, because every component intersects <math>S^*</math>, all components must be contained in a single component, hence the coarse space <math>H_g</math> is irreducible. From the general theory of algebraic stacks, this implies the stack quotient <math>\mathcal{M}_g</math> is irreducible.

=== Properness ===
[[Proper scheme|Properness]], or [[Compact space|compactness]] for [[orbifold]]s, follows from a theorem on stable reduction on curves.<ref name=":0" /> This can be found using a theorem of [[Alexander Grothendieck|Grothendieck]] regarding the stable reduction of [[Abelian variety|Abelian varieties]], and showing its equivalence to the stable reduction of curves.<ref name=":0" /><sup>section 5.2</sup>

===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 <math>g > 1</math>; 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 <math>0</math> curves can be established using [[deformation Theory]]. The number of moduli for a genus <math>0</math> curve, e.g. <math>\mathbb{P}^1</math>, is given by the cohomology group<blockquote><math>H^1(C,T_C)</math></blockquote>With [[Serre duality]] this cohomology group is isomorphic to<blockquote><math>\begin{align}
H^1(C,T_C) &\cong H^0(C, \omega_C\otimes T_C^\vee) \\
&\cong H^0(C, \omega_C^{\otimes 2})

\end{align}</math></blockquote>for the dualizing sheaf <math>\omega_C</math>. But, using [[Riemann–Roch theorem|Riemann–Roch]] shows the degree of the canonical bundle is <math>-2</math>, so the degree of <math>\omega_C^{\otimes 2}</math> is <math>-4</math>, hence there are no global sections, meaning<blockquote><math>H^0(C,\omega_C^{\otimes 2}) = 0</math></blockquote>showing there are no deformations of genus <math>0</math> curves. This proves <math>\mathcal{M}_0</math> is just a single point, and the only genus <math>0</math> curves is given by <math>\mathbb{P}^1</math>. The only technical difficulty is the automorphism group of <math>\mathbb{P}^1</math> is the [[algebraic group]] <math>\text{PGL}(2,\mathbb{C})</math>, which rigidifies once three points<ref name=":2" /> on <math>\mathbb{P}^1</math> are fixed, so most authors take <math>\mathcal{M}_0</math> to mean <math>\mathcal{M}_{0,3}</math>.

=== Genus 1 ===
((Main|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]]<blockquote><math>j: \mathcal{M}_{1,1}|_{\mathbb{C)) \to \mathbb{A}^1_\mathbb{C}</math></blockquote>where <math>\mathcal{M}_{1,1}|_{\mathbb{C))=\mathcal{M}_{1,1}\times_{\text{Spec}(\mathbb{Z})} \text{Spec}(\mathbb{C})</math>. Topologically, <math>\mathcal{M}_{1,1}|_{\mathbb{C))</math> is just the affine line, but it can be compactified to a stack with underlying topological space <math>\mathbb{P}^1_\mathbb{C}</math> by adding a stable curve at infinity. This is an elliptic curve with a single cusp. The construction of the general case over <math>\text{Spec}(\mathbb{Z})</math> was originally completed by [[Pierre Deligne|Deligne]] and [[Michael Rapoport|Rapoport]].<ref>((Citation|last1=Deligne|first1=P.|title=Les schémas de modules de courbes elliptiques|pages=143–316|publisher=Springer Berlin Heidelberg|isbn=978-3-540-06558-6|last2=Rapoport|first2=M.|series=Lecture Notes in Mathematics|year=1973|volume=349|doi=10.1007/bfb0066716)), URL: http://publications.ias.edu/node/367</ref>

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 <math>\mathcal{M}_1</math> would have stabilizer group at the point <math>[C] \in \mathcal{M}_1</math> 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, <math>\mathcal{M}_{1,1}</math> is a smooth [[Deligne–Mumford stack]].

=== Genus 2 ===

==== Affine parameter space ====
In genus 2 it is a [[Linear system of divisors#Hyperelliptic curves|classical result]] that all such curves are [[Hyperelliptic curve|hyperelliptic]],<ref>((Cite book|last=Hartshorne| first=Robin|author-link=Robin Hartshorne|title=Algebraic geometry| date=29 June 2013|isbn=978-1-4757-3849-0|location=New York|oclc=861706007))</ref><sup>pg 298</sup> 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

:<math>y^2 - x(x-1)(x-a)(x-b)(x-c)</math>

for some uniquely defined <math>a,b,c \in \mathbb{A}^1</math>, the parameter space for such curves is given by

:<math>\mathbb{A}^3 \setminus (\Delta_{a,b} \cup \Delta_{a,c} \cup \Delta_{b,c}),</math>

where <math>\Delta_{i,j}</math> corresponds to the locus <math>i \neq j</math>.<ref>((Cite journal|last=Igusa|first=Jun-Ichi|author-link=Jun-Ichi Igusa|date=1960|title=Arithmetic Variety of Moduli for Genus Two|jstor=1970233|journal=[[Annals of Mathematics]]|volume=72|issue=3|pages=612–649|doi=10.2307/1970233|issn=0003-486X))</ref>

==== 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<ref>((Cite arXiv|last=Larson|first=Eric|date=2019-04-17|title=The integral Chow ring of <math>\overline{M}_2</math>|class=math.AG| eprint=1904.08081))</ref>

:<math>z^2 = ax^6 + bx^5y + cx^4y^2 + dx^3y^3 + ex^2y^4 + fxy^5 + gy^6 ,</math>

where <math>a,\ldots,f</math> are parameters for sections of <math>\Gamma(\mathbb{P}(3,1), \mathcal{O}(g))</math>. Then, the locus of sections which contain no triple root contains every curve <math>C</math> represented by a point <math>[C]\in \mathcal{M}_2</math>.

=== Genus 3 ===
This is the first moduli space of curves which has both a hyperelliptic locus and a non-hyperelliptic locus.<ref>((Citation|last1=Girard|first1=Martine|title=Classification of Genus 3 Curves in Special Strata of the Moduli Space| date=2006|work=Algorithmic Number Theory|volume=4076|pages=346–360|editor-last=Hess|editor-first=Florian|place=Berlin, Heidelberg|publisher=Springer Berlin Heidelberg|doi=10.1007/11792086_25|isbn=978-3-540-36075-9|last2=Kohel|first2=David R.|editor2-last=Pauli|editor2-first=Sebastian|editor3-last=Pohst|editor3-first=Michael|arxiv=math/0603555|bibcode=2006math......3555G|mr=2282935|s2cid=15638167))</ref><ref>((cite journal | last1=Penev | first1=Nikola | last2=Vakil | first2=Ravi | author2-link=Ravi Vakil|title=The Chow ring of the moduli space of curves of genus six | journal=[[Algebraic Geometry (journal)|Algebraic Geometry]] | volume=2 | issue=1 | year=2015 | issn=2214-2584 | doi=10.14231/ag-2015-006 | pages=123–136|mr=3322200|arxiv=1307.6614| s2cid=54876684 ))</ref> 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

:<math>\operatorname{Hilb}_{\mathbb{P}^2}^{8t-4} \cong \mathbb{P}^{\binom{6}{4} - 1}</math>.

Then, the moduli space is stratified by the substacks

:<math>\mathcal{M}_3 = [H_2/\mathrm{PGL}(3))] \coprod \mathcal{M}_3^{\mathrm{hyp))</math>.

== 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<blockquote><math>\mathbb{P}^n \to \mathcal{M}_g</math></blockquote>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 <math>10</math>.<ref>((Cite book|last=Severi, Francesco, 1879-1961.|title=Sulla classificazione delle curve algebriche e sul teorema d'esistenza di Riemann|date=1915|publisher=Tipografia della R. Accademia dei Lincei|oclc=881814709))</ref> Although, it turns out that for genus <math>g \geq 23</math><ref>((Cite journal|last1=Eisenbud|first1=David|author1-link=David Eisenbud|last2=Harris|first2=Joe|author2-link=Joe Harris (mathematician)| year=1987|title=The Kodaira dimension of the moduli space of curves of genus ?23|journal=[[Inventiones Mathematicae]]|volume=90|issue=2|pages=359–387|doi=10.1007/bf01388710|bibcode=1987InMat..90..359E|s2cid=120642775|issn=0020-9910))</ref><ref>((Citation|last1=Harris|first1=Joe|author1-link=Joe Harris (mathematician)|title=On the Kodaira Dimension of the Moduli Space of Curves|date=1982|work=Selected Papers|pages=171–234|place=New York, NY|publisher=Springer New York|isbn=978-1-4419-1936-6|last2=Mumford|first2=David|doi=10.1007/978-1-4757-4265-7_8|url=https://dash.harvard.edu/bitstream/1/3613574/3/Mumford_OnKodairaDim.pdf |author2-link=David Mumford))</ref><ref>((Citation|last1=Harris|first1=Joe|title=On the Kodaira Dimension of the Moduli Space of Curves|date=1982|work=Selected Papers|pages=171–234|place=New York, NY|publisher=Springer New York|isbn=978-1-4419-1936-6|last2=Mumford|first2=David|doi=10.1007/978-1-4757-4265-7_8|url=http://nrs.harvard.edu/urn-3:HUL.InstRepos:3613574 ))</ref> 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
:<math>\kappa_g = \mathrm{Kod}(\overline{\mathcal{M))_{g}),</math>
and found <math>\kappa_g > 0 </math> for <math>g \geq 23</math>.  In fact, for <math>g > 23</math>,
:<math>\kappa_g = 3g - 3 = \dim(\mathcal{M}_g),</math>
and hence <math>\mathcal{M}_g</math> 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 <math>\mathcal{C}_g</math>.<ref>((cite book|s2cid=8281102 |doi=10.1090/pspum/080.1/2483934 |chapter=The global geometry of the moduli space of curves |title=Algebraic Geometry |series=Proceedings of Symposia in Pure Mathematics |year=2009 |last1=Farkas |first1=Gavril |volume=80 |issue=1 |pages=125–147 |isbn=9780821847022 ))</ref>

== Stratification of boundary ==
The moduli space <math>\overline{\mathcal{M))_{g}</math> has a natural stratification on the boundary <math>\partial\overline{\mathcal{M))_{g}</math> whose points represent singular genus <math>g</math> curves.<ref name=":1">((Cite book|title=Arithmetic and geometry: papers dedicated to I.R. Shafarevich on the occasion of his sixtieth birthday|date=1983|publisher=Birkhäuser|others=Shafarevich, Igor Rostislavovich, 1923-2017, Artin, Michael, Tate, John Torrence, 1925-2019|isbn=978-1-4757-9286-7| location=Boston|oclc=681426064|url=https://www.dam.brown.edu/people/mumford/alg_geom/papers/1983b--EnumGeomModuli-NC.pdf))</ref> It decomposes into strata

:<math>\partial\overline{\mathcal{M))_{g} = \coprod_{0 \leq h \leq (g/2)} \Delta_h^*</math>,

where

* <math>\Delta_h^* \cong \overline{\mathcal{M))_{h} \times \overline{\mathcal{M))_{g-h}  </math> for <math>1 \leq h < g/2</math>.
* <math>\Delta_0^* \cong \overline{\mathcal{M))_{g-1,2} / (\Z/2)</math> where the action permutes the two marked points.
* <math>\Delta_{g/2} \cong (\overline{\mathcal{M))_{g/2} \times \overline{\mathcal{M))_{g/2}) / (\Z/2)</math> whenever <math>g</math> is even.

The curves lying above these loci correspond to

* A pair of curves <math>C, C'</math> connected at a double point.
* The [[Normal scheme|normalization]] of a genus <math>g</math> 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 <math>2</math> case, there is a stratification given by

:<math>\begin{align}
\partial \overline{\mathcal{M))_2 &= \Delta_0^* \coprod \Delta_1^* \\
&= \overline{\mathcal{M))_{1,2}/(\Z/2)
\coprod
(\overline{\mathcal{M))_1\times \overline{\mathcal{M))_1)/(\Z/2)
\end{align}</math>.

Further analysis of these strata can be used to give the generators of the [[Chow ring]] <math>A^*(\overline{\mathcal{M))_2)</math><ref name=":1" /> <sup>proposition 9.1</sup>.

==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 <math>\mathcal{M}_{g,n}</math> (or <math>\overline{\mathcal{M))_{g,n}</math>), and have dimension <math>3g-3 + n</math>.

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 [[Moduli stack of elliptic curves|stack of elliptic curves]]. Level 1 [[modular form]]s 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 structure (algebraic geometry)|level ''N'' structure]] (roughly a marking of the points of order ''N'').

==Boundary geometry==
An important property of the compactified moduli spaces <math>\overline{\mathcal{M))_{g,n}</math> is that their boundary can be described in terms of moduli spaces <math>\overline{\mathcal{M))_{g',n'}</math> for genera <math>g' < g</math>.  Given a marked, stable, nodal curve one can associate its ''dual graph'', a [[Graph (discrete mathematics)|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 component]]s 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 <math>\overline{\mathcal{M))_{g,n}</math> is isomorphic to the [[stack quotient]] of a product <math>\prod_v \overline{\mathcal{M))_{g_v,n_v}</math> of compactified moduli spaces of curves by a finite group. In the product the factor corresponding to a vertex ''v'' has genus g<sub>v</sub> taken from the labelling and number of markings <math>n_v</math> equal to the number of outgoing edges and half-edges at ''v''.  The total genus ''g'' is the sum of the g<sub>v</sub> plus the number of closed cycles in the graph.

Stable curves whose dual graph contains a vertex labelled by <math>g_v=g</math> (hence all other vertices have <math>g_v=0</math> and the graph is a tree) are called "rational tail" and their moduli space is denoted <math>\mathcal{M}^{\mathrm{r.t.))_{g,n}</math>.  Stable curves whose dual graph is a tree are called "compact type" (because the Jacobian is compact) and their moduli space is denoted <math>\mathcal{M}^{\mathrm{c.))_{g,n}</math>.<ref name=":2">((cite arXiv |eprint=1101.5489|last1= Faber|first1= Carel|title= Tautological and non-tautological cohomology of the moduli space of curves |last2=  Pandharipande|first2= Rahul |author-link2=Rahul Pandharipande |class= math.AG|year= 2011))</ref>

==See also==
*[[Moduli of marked curves]]
*[[Witten conjecture]]
*[[Tautological ring]]
*[[Grothendieck–Riemann–Roch theorem]]

==References==
((reflist))

=== Classic references ===
*((Cite journal
  | last = Grothendieck
  | first = Alexander
  | author-link = Alexander Grothendieck
  | title = Techniques de construction en géométrie analytique. I. Description axiomatique de l'espace de Teichmüller et de ses variantes
  | place = Paris
  | year = 1960–1961
  | journal = Séminaire Henri Cartan |volume=13 |issue= 1|id= Exposés No. 7 and 8
  | url = http://archive.numdam.org/article/SHC_1960-1961__13_1_A4_0.pdf
|zbl=0142.33503    ))

*((Cite book|last1=Mumford| first1=David|author1-link=David Mumford|title=Geometric invariant theory|date=1994|publisher=Springer-Verlag|last2=Fogarty| first2=John|last3=Kirwan|first3= Frances Clare|author3-link=Frances Kirwan|edition=3rd enl. |location=Berlin|oclc=29184987|isbn=3-540-56963-4|mr=1304906))
*((cite journal
  | last1 = Deligne|first1 = Pierre |author1-link = Pierre Deligne
  | last2=Mumford|first2= David |author2-link=David Mumford
  | title = The irreducibility of the space of curves of given genus
  | journal = [[Publications Mathématiques de l'IHÉS]]
  | volume = 36
  | pages = 75–109
  | date = 1969
  | url = http://archive.numdam.org/article/PMIHES_1969__36__75_0.pdf
  | doi = 10.1007/bf02684599
  | citeseerx = 10.1.1.589.288
  |s2cid = 16482150 ))

=== Books on moduli of curves ===
*((cite book
  | last = Harris
  | first = Joe
  | author-link = Joe Harris (mathematician)
  | author2=Morrison, Ian
  | title = Moduli of Curves
  | publisher = [[Springer Verlag]]
  | date = 1998
  | isbn =978-0-387-98429-2 ))

*((cite book
  | last1 = Katz
  | first1 = Nicholas M
  | author-link = Nick Katz
  | last2=Mazur|first2= Barry |author-link2=Barry Mazur
  | title = Arithmetic Moduli of Elliptic Curves
  | publisher = [[Princeton University Press]]
  | date = 1985
  | isbn =978-0-691-08352-0 ))
* ((cite book | last1 =  Arbarello| first1 = Enrico| last2 =  Cornalba| first2 = Maurizio | last3 = Griffiths| first3 = Phillip A.|doi=10.1007/978-3-540-69392-5|title=Geometry of Algebraic Curves II|series=Grundlehren der mathematischen Wissenschaften |year=2011 |volume=268 |isbn=978-3-540-42688-2 ))

=== Cohomology and intersection theory ===
*((cite book|last=Zvonkine|first=Dimitri|chapter=An introduction to moduli spaces of curves and their intersection theory|year=2012|title=Handbook of Teichmüller Theory, Volume III|series=IRMA Lectures in Mathematics and Theoretical Physics |volume=17 |pages=667–716|editor-last=Papadopoulos|editor-first=Athanase|place=Zürich, Switzerland|publisher=European Mathematical Society Publishing House|url=https://www-fourier.ujf-grenoble.fr/sites/ifmaquette.ujf-grenoble.fr/files/ete2011-zvonkine.pdf|doi=10.4171/103-1/12|isbn=978-3-03719-103-3|mr=2952773))
* ((cite book|url=https://people.math.ethz.ch/~rahul/tnt.pdf|title= Tautological and non-tautological cohomology of the moduli space of curves|first1= Carel |last1=Faber|first2=Rahul |last2=Pandharipande|author2-link= Rahul Pandharipande|work=Handbook of moduli. Vol. I|pages=293–330
|series=Advanced Lectures in Mathematics (ALM)|volume= 24|editor1-first=Gavril |editor1-last=Farkas|editor2-first=Ian |editor2-last= Morrison |publisher=International Press|location= Somerville, MA|year=2013|isbn=9781571462572|mr=3184167))

==External links==
*((cite web |title=Topology and geometry of the moduli space of curves |url=https://aimath.org/WWN/modspacecurves/ |website=The American Institute of Mathematics))
* [https://web.archive.org/web/20180123190525/http://w3.impa.br/~massaren/files/smgwqc.pdf "Moduli of Stable Maps, Gromov-Witten Invariants, and Quantum Cohomology"]
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