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").
Moduli stacks of stable curves
[edit]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 (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 . 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(2). 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
[edit]It is a non-trivial theorem, proved by Pierre Deligne and David Mumford,[1] 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 (from the embedding of the very ample for every curve) which have Hilbert polynomial . Then, the stack is a construction of the moduli space . Using deformation theory, 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 as
- ,
where is the subscheme of smooth stable curves and is an irreducible component of . They analyze the components of (as a GIT quotient). 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle S^*} , 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
[edit]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
[edit]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
[edit]Genus 0
[edit]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
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[2] on are fixed, so most authors take to mean .
Genus 1
[edit]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.[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 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
[edit]Affine parameter space
[edit]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
for some uniquely defined , the parameter space for such curves is given by
where corresponds to the locus .[5]
Weighted projective space
[edit]Using a weighted projective space and the Riemann–Hurwitz formula, a hyperelliptic curve can be described as a polynomial of the form[6]
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
[edit]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
- .
Then, the moduli space is stratified by the substacks
- .
Birational geometry
[edit]Unirationality conjecture
[edit]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 had proved this to be true for genera up to .[9] Although, it turns out that for genus [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
and found for . In fact, for ,
and hence is of general type.
Geometric implication
[edit]This is significant geometrically because it implies any linear system on a ruled variety cannot contain the universal curve .[13]
Stratification of boundary
[edit]The moduli space has a natural stratification on the boundary whose points represent singular genus curves.[14] It decomposes into strata
- ,
where
- for .
- where the action permutes the two marked points.
- whenever is even.
The curves lying above these loci correspond to
- A pair of curves connected at a double point.
- The normalization of a genus 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
[edit]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 [14] proposition 9.1.
Moduli of marked curves
[edit]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 (or ), 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 (roughly a marking of the points of order N).
Boundary geometry
[edit]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 (hence all other vertices have and the graph is a tree) are called "rational tail" and their moduli space is denoted . Stable curves whose dual graph is a tree are called "compact type" (because the Jacobian is compact) and their moduli space is denoted .[2]
See also
[edit]References
[edit]- ^ a b c Deligne, Pierre; Mumford, David (1969). "The irreducibility of the space of curves of given genus". Publications Mathématiques de l'IHÉS. 36: 75–109. doi:10.1007/BF02684599. S2CID 16482150.
- ^ a b Faber, Carel; Pandharipande, Rahul (2011). "Tautological and non-tautological cohomology of the moduli space of curves". arXiv:1101.5489 [math.AG].
- ^ Deligne, P.; Rapoport, M. (1973), Les schémas de modules de courbes elliptiques, Lecture Notes in Mathematics, vol. 349, Springer Berlin Heidelberg, pp. 143–316, doi:10.1007/bfb0066716, ISBN 978-3-540-06558-6, URL: http://publications.ias.edu/node/367
- ^ Hartshorne, Robin (29 June 2013). Algebraic geometry. New York. ISBN 978-1-4757-3849-0. OCLC 861706007.
{{cite book}}
: CS1 maint: location missing publisher (link) - ^ 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. JSTOR 1970233.
- ^ Larson, Eric (2019-04-17). "The integral Chow ring of ". arXiv:1904.08081 [math.AG].
- ^ 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, vol. 4076, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 346–360, arXiv:math/0603555, Bibcode:2006math......3555G, doi:10.1007/11792086_25, ISBN 978-3-540-36075-9, MR 2282935, S2CID 15638167
- ^ Penev, Nikola; Vakil, Ravi (2015). "The Chow ring of the moduli space of curves of genus six". Algebraic Geometry. 2 (1): 123–136. arXiv:1307.6614. doi:10.14231/ag-2015-006. ISSN 2214-2584. MR 3322200. S2CID 54876684.
- ^ 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.
{{cite book}}
: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) - ^ Eisenbud, David; Harris, Joe (1987). "The Kodaira dimension of the moduli space of curves of genus ?23". Inventiones Mathematicae. 90 (2): 359–387. Bibcode:1987InMat..90..359E. doi:10.1007/bf01388710. ISSN 0020-9910. S2CID 120642775.
- ^ Harris, Joe; Mumford, David (1982), "On the Kodaira Dimension of the Moduli Space of Curves" (PDF), 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
- ^ 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
- ^ Farkas, Gavril (2009). "The global geometry of the moduli space of curves". Algebraic Geometry. Proceedings of Symposia in Pure Mathematics. Vol. 80. pp. 125–147. doi:10.1090/pspum/080.1/2483934. ISBN 9780821847022. S2CID 8281102.
- ^ a b Arithmetic and geometry: papers dedicated to I.R. Shafarevich on the occasion of his sixtieth birthday (PDF). Shafarevich, Igor Rostislavovich, 1923-2017, Artin, Michael, Tate, John Torrence, 1925-2019. Boston: Birkhäuser. 1983. ISBN 978-1-4757-9286-7. OCLC 681426064.
{{cite book}}
: CS1 maint: others (link)
Classic references
[edit]- 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. 13 (1). Paris. 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.
Books on moduli of curves
[edit]- 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.
Cohomology and intersection theory
[edit]- 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). "Tautological and non-tautological cohomology of the moduli space of curves" (PDF). In Farkas, Gavril; Morrison, Ian (eds.). Handbook of Moduli, Vol. I. Advanced Lectures in Mathematics (ALM). Vol. 24. Somerville, MA: International Press. pp. 293–330. ISBN 9781571462572. MR 3184167.
External links
[edit]- "Topology and geometry of the moduli space of curves". The American Institute of Mathematics.
- "Moduli of Stable Maps, Gromov-Witten Invariants, and Quantum Cohomology"