Moduli of algebraic curves

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

${\displaystyle H_{g}^{o}\coprod H_{g,1}\coprod \cdots \coprod H_{g,n}}$,

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

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}

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

${\displaystyle H^{1}(C,T_{C})}$

With Serre duality this cohomology group is isomorphic to

${\displaystyle {\begin{aligned}H^{1}(C,T_{C})&\cong H^{0}(C,\omega _{C}\otimes T_{C}^{\vee })\\&\cong H^{0}(C,\omega _{C}^{\otimes 2})\end{aligned}}}$

for the dualizing sheaf ${\displaystyle \omega _{C}}$. But, using Riemann–Roch shows the degree of the canonical bundle is ${\displaystyle -2}$, so the degree of ${\displaystyle \omega _{C}^{\otimes 2}}$ is ${\displaystyle -4}$, hence there are no global sections, meaning

${\displaystyle H^{0}(C,\omega _{C}^{\otimes 2})=0}$

showing there are no deformations of genus ${\displaystyle 0}$ curves. This proves ${\displaystyle {\mathcal {M}}_{0}}$ is just a single point, and the only genus ${\displaystyle 0}$ curves is given by ${\displaystyle \mathbb {P} ^{1}}$. The only technical difficulty is the automorphism group of ${\displaystyle \mathbb {P} ^{1}}$ is the algebraic group ${\displaystyle {\text{PGL}}(2,\mathbb {C} )}$, which rigidifies once three points[2] on ${\displaystyle \mathbb {P} ^{1}}$ are fixed, so most authors take ${\displaystyle {\mathcal {M}}_{0}}$ to mean ${\displaystyle {\mathcal {M}}_{0,3}}$.

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

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

where ${\displaystyle {\mathcal {M}}_{1,1}|_{\mathbb {C} }={\mathcal {M}}_{1,1}\times _{{\text{Spec}}(\mathbb {Z} )}{\text{Spec}}(\mathbb {C} )}$. Topologically, ${\displaystyle {\mathcal {M}}_{1,1}|_{\mathbb {C} }}$ is just the affine line, but it can be compactified to a stack with underlying topological space ${\displaystyle \mathbb {P} _{\mathbb {C} }^{1}}$ by adding a stable curve at infinity. This is an elliptic curve with a single cusp. The construction of the general case over ${\displaystyle {\text{Spec}}(\mathbb {Z} )}$ 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 ${\displaystyle {\mathcal {M}}_{1}}$ would have stabilizer group at the point ${\displaystyle [C]\in {\mathcal {M}}_{1}}$ 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, ${\displaystyle {\mathcal {M}}_{1,1}}$ is a smooth Deligne–Mumford stack.

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

${\displaystyle y^{2}-x(x-1)(x-a)(x-b)(x-c)}$

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

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

where ${\displaystyle \Delta _{i,j}}$ corresponds to the locus ${\displaystyle i\neq j}$.[5]

Using a weighted projective space and the Riemann–Hurwitz formula, a hyperelliptic curve can be described as a polynomial of the form[6]

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

where ${\displaystyle a,\ldots ,f}$ are parameters for sections of ${\displaystyle \Gamma (\mathbb {P} (3,1),{\mathcal {O}}(g))}$. Then, the locus of sections which contain no triple root contains every curve ${\displaystyle C}$ represented by a point ${\displaystyle [C]\in {\mathcal {M}}_{2}}$.

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

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

Then, the moduli space is stratified by the substacks

${\displaystyle {\mathcal {M}}_{3}=[H_{2}/\mathrm {PGL} (3))]\coprod {\mathcal {M}}_{3}^{\mathrm {hyp} }}$.

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

${\displaystyle \mathbb {P} ^{n}\to {\mathcal {M}}_{g}}$

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 ${\displaystyle 10}$.[9] Although, it turns out that for genus ${\displaystyle g\geq 23}$[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

${\displaystyle \kappa _{g}=\mathrm {Kod} ({\overline {\mathcal {M}}}_{g}),}$

and found ${\displaystyle \kappa _{g}>0}$ for ${\displaystyle g\geq 23}$. In fact, for ${\displaystyle g>23}$,

${\displaystyle \kappa _{g}=3g-3=\dim({\mathcal {M}}_{g}),}$

and hence ${\displaystyle {\mathcal {M}}_{g}}$ is of general type.

This is significant geometrically because it implies any linear system on a ruled variety cannot contain the universal curve ${\displaystyle {\mathcal {C}}_{g}}$.[13]

For the genus ${\displaystyle 2}$ case, there is a stratification given by

${\displaystyle {\begin{aligned}\partial {\overline {\mathcal {M}}}_{2}&=\Delta _{0}^{*}\coprod \Delta _{1}^{*}\\&={\overline {\mathcal {M}}}_{1,2}/(\mathbb {Z} /2)\coprod ({\overline {\mathcal {M}}}_{1}\times {\overline {\mathcal {M}}}_{1})/(\mathbb {Z} /2)\end{aligned}}}$.

Further analysis of these strata can be used to give the generators of the Chow ring ${\displaystyle A^{*}({\overline {\mathcal {M}}}_{2})}$[14] ^{proposition 9.1}.

• Grothendieck, Alexander (1960–1961). "Techniques de construction en géométrie analytique. I. Description axiomatique de l'espace de Teichmüller et de ses variantes" (PDF). Séminaire Henri Cartan. Paris. 13 (1). Zbl 0142.33503. Exposés No. 7 and 8.

• Mumford, David; Fogarty, John; Kirwan, Frances Clare (1994). Geometric invariant theory (3rd enl. ed.). Berlin: Springer-Verlag. ISBN 3-540-56963-4. MR 1304906. OCLC 29184987.
• Deligne, Pierre; Mumford, David (1969). "The irreducibility of the space of curves of given genus" (PDF). Publications Mathématiques de l'IHÉS. 36: 75–109. CiteSeerX 10.1.1.589.288. doi:10.1007/bf02684599. S2CID 16482150.

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

• Katz, Nicholas M; Mazur, Barry (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. Vol. 268. doi:10.1007/978-3-540-69392-5. ISBN 978-3-540-42688-2.

• Zvonkine, Dimitri (2012). "An introduction to moduli spaces of curves and their intersection theory". In Papadopoulos, Athanase (ed.). Handbook of Teichmüller Theory, Volume III (PDF). IRMA Lectures in Mathematics and Theoretical Physics. Vol. 17. Zürich, Switzerland: European Mathematical Society Publishing House. pp. 667–716. doi:10.4171/103-1/12. ISBN 978-3-03719-103-3. MR 2952773.
• Faber, Carel; Pandharipande, Rahul (2013). Farkas, Gavril; Morrison, Ian (eds.). Tautological and non-tautological cohomology of the moduli space of curves (PDF). pp. 293–330. ISBN 9781571462572. MR 3184167. {{cite book}}: |work= ignored (help)