moduli of algebraic curves

The moduli stack $\backslash mathcal\{M\}\_\{g\}$ 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 $\backslash overline\{\backslash mathcal\{M\}\}\_\{g\}$. 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 $\backslash mathcal\{M\}\_0$ is equal to

:

&= -3 .\end{align}

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 $\backslash mathcal\{M\}\_1$ has dimension 0.

It is a non-trivial theorem, proved by Pierre Deligne and David Mumford,{{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}} that the moduli stack $\backslash mathcal\{M\}\_g$ is irreducible, meaning it cannot be expressed as the union of two proper substacks. They prove this by analyzing the locus $H\_g$ of stable curves in the Hilbert scheme $\backslash mathrm\{Hilb\}\_\{\backslash mathbb\{P\}^\{5g\; -\; 5\; -1\}\}^\{P\_g(n)\}$ of tri-canonically embedded curves (from the embedding of the very ample $\backslash omega\_C^\{\backslash otimes\; 3\}$ for every curve) which have Hilbert polynomial $P\_g(n)\; =\; (6n-1)(g-1)$. Then, the stack $[H\_g\; /\; \backslash mathrm\{PGL\}(5g-6)]$ is a construction of the moduli space $\backslash mathcal\{M\}\_g$. Using deformation theory, Deligne and Mumford show this stack is smooth and use the stack of isomorphisms between stable curves $\backslash mathrm\{Isom\}\_S(C,C\text{'})$, to show that $\backslash mathcal\{M\}\_g$ has finite stabilizers, hence it is a Deligne–Mumford stack. Moreover, they find a stratification of $H\_g$ as

:$H\_g^o\; \backslash coprod\; H\_\{g,1\}\; \backslash coprod\; \backslash cdots\; \backslash coprod\; H\_\{g,n\}$,

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

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.

Determining the geometry of the moduli space of genus $0$ curves can be established by using deformation Theory. The number of moduli for a genus $0$ curve, e.g. $\backslash mathbb\{P\}^1$, is given by the cohomology group

$H^1(C,T\_C)$

$\backslash begin\{align\}$

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}

$H^0(C,\backslash omega\_C^\{\backslash otimes\; 2\})\; =\; 0$

{{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

$j:\; \backslash mathcal\{M\}\_\{1,1\}|\_\{\backslash mathbb\{C\}\}\; \backslash to\; \backslash mathbb\{A\}^1\_\backslash mathbb\{C\}$

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

In genus 2 it is a classical result that all such curves are hyperelliptic,{{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}}^{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

:$y^2\; -\; x(x-1)(x-a)(x-b)(x-c)$

for some uniquely defined $a,b,c\; \backslash in\; \backslash mathbb\{A\}^1$, the parameter space for such curves is given by

:$\backslash mathbb\{A\}^3\; \backslash setminus\; (\backslash Delta\_\{a,b\}\; \backslash cup\; \backslash Delta\_\{a,c\}\; \backslash cup\; \backslash Delta\_\{b,c\}),$

where $\backslash Delta\_\{i,j\}$ corresponds to the locus $i\; \backslash neq\; j$.{{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}}

Using a weighted projective space and the Riemann–Hurwitz formula, a hyperelliptic curve can be described as a polynomial of the form{{Cite arXiv|last=Larson|first=Eric|date=2019-04-17|title=The integral Chow ring of $\backslash overline\{M\}\_2$|class=math.AG| eprint=1904.08081}}

:$z^2\; =\; ax^6\; +\; bx^5y\; +\; cx^4y^2\; +\; dx^3y^3\; +\; ex^2y^4\; +\; fxy^5\; +\; gy^6\; ,$

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

This is the first moduli space of curves which has both a hyperelliptic locus and a non-hyperelliptic locus.{{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}}{{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 | volume=2 | issue=1 | year=2015 | issn=2214-2584 | doi=10.14231/ag-2015-006 | pages=123–136| doi-access=free |mr=3322200|arxiv=1307.6614| s2cid=54876684 }} 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

:$\backslash operatorname\{Hilb\}\_\{\backslash mathbb\{P\}^2\}^\{8t-4\}\; \backslash cong\; \backslash mathbb\{P\}^\{\backslash binom\{6\}\{4\}\; -\; 1\}$.

Then, the moduli space is stratified by the substacks

:$\backslash mathcal\{M\}\_3\; =\; [H\_2/\backslash mathrm\{PGL\}(3))]\; \backslash coprod\; \backslash mathcal\{M\}\_3^\{\backslash mathrm\{hyp\}\}$.

In all of the previous cases, the moduli spaces can be found to be unirational, meaning there exists a dominant rational morphism

$\backslash mathbb\{P\}^n\; \backslash to\; \backslash mathcal\{M\}\_g$

:$\backslash kappa\_g\; =\; \backslash mathrm\{Kod\}(\backslash overline\{\backslash mathcal\{M\}\}\_\{g\}),$

and found $\backslash kappa\_g\; >\; 0$ for $g\; \backslash geq\; 23$. In fact, for $g\; >\; 23$,

:$\backslash kappa\_g\; =\; 3g\; -\; 3\; =\; \backslash dim(\backslash mathcal\{M\}\_g),$

and hence $\backslash mathcal\{M\}\_g$ is of general type.

This is significant geometrically because it implies any linear system on a ruled variety cannot contain the universal curve $\backslash mathcal\{C\}\_g$.{{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 |last=Farkas |first=Gavril|author-link=Gavril Farkas| volume=80 |issue=1 |pages=125–147 |isbn=9780821847022 }}

The moduli space $\backslash overline\{\backslash mathcal\{M\}\}\_\{g\}$ has a natural stratification on the boundary $\backslash partial\backslash overline\{\backslash mathcal\{M\}\}\_\{g\}$ whose points represent singular genus $g$ curves.{{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}} It decomposes into strata

:$\backslash partial\backslash overline\{\backslash mathcal\{M\}\}\_\{g\}\; =\; \backslash coprod\_\{0\; \backslash leq\; h\; \backslash leq\; (g/2)\}\; \backslash Delta\_h^*$,

where

• $\backslash Delta\_h^*\; \backslash cong\; \backslash overline\{\backslash mathcal\{M\}\}\_\{h\}\; \backslash times\; \backslash overline\{\backslash mathcal\{M\}\}\_\{g-h\}$ for .
• $\backslash Delta\_0^*\; \backslash cong\; \backslash overline\{\backslash mathcal\{M\}\}\_\{g-1,2\}\; /\; (\backslash Z/2)$ where the action permutes the two marked points.
• $\backslash Delta\_\{g/2\}\; \backslash cong\; (\backslash overline\{\backslash mathcal\{M\}\}\_\{g/2\}\; \backslash times\; \backslash overline\{\backslash mathcal\{M\}\}\_\{g/2\})\; /\; (\backslash Z/2)$ whenever $g$ is even.

The curves lying above these loci correspond to

• A pair of curves $C,\; C\text{'}$ 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.

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

:$\backslash begin\{align\}$

\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}.

Further analysis of these strata can be used to give the generators of the Chow ring $A^*(\backslash overline\{\backslash mathcal\{M\}\}\_2)$ ^{proposition 9.1}.

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 $\backslash mathcal\{M\}\_\{g,n\}$ (or $\backslash overline\{\backslash mathcal\{M\}\}\_\{g,n\}$), and have dimension $3g-3\; +\; n$.

A case of particular interest is the moduli stack $\backslash overline\{\backslash mathcal\{M\}\}\_\{1,1\}$ 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).

An important property of the compactified moduli spaces $\backslash overline\{\backslash mathcal\{M\}\}\_\{g,n\}$ is that their boundary can be described in terms of moduli spaces $\backslash overline\{\backslash mathcal\{M\}\}\_\{g\text{'},n\text{'}\}$ 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 $\backslash overline\{\backslash mathcal\{M\}\}\_\{g,n\}$ is isomorphic to the stack quotient of a product $\backslash prod\_v\; \backslash overline\{\backslash mathcal\{M\}\}\_\{g\_v,n\_v\}$ of compactified moduli spaces of curves by a finite group. In the product the factor corresponding to a vertex v has genus g_v taken from the labelling and number of markings $n\_v$ equal to the number of outgoing edges and half-edges at v. The total genus g is the sum of the g_v plus the number of closed cycles in the graph.

Stable curves whose dual graph contains a vertex labelled by $g\_v=g$ (hence all other vertices have $g\_v=0$ and the graph is a tree) are called "rational tail" and their moduli space is denoted $\backslash mathcal\{M\}^\{\backslash mathrm\{r.t.\}\}\_\{g,n\}$. Stable curves whose dual graph is a tree are called "compact type" (because the Jacobian is compact) and their moduli space is denoted $\backslash mathcal\{M\}^\{\backslash mathrm\{c.\}\}\_\{g,n\}$.{{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}}

• Witten conjecture
• Tautological ring
• Grothendieck–Riemann–Roch theorem

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|series=Advanced Lectures in Mathematics (ALM)|volume= 24|editor1-first=Gavril |editor1-last=Farkas|editor1-link=Gavril Farkas|editor2-first=Ian |editor2-last= Morrison |publisher=International Press|location= Somerville, MA|year=2013|isbn=9781571462572|mr=3184167}}

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