Math 2.0

Algebraic geometry Definition

Moduli space

Builds on SchemeFunctorFunctor of pointsRepresentable functorAlgebraically closed field

Definition

Let SS be a base scheme, and let F ⁣:(Sch/S)op→SetF \colon (\mathrm{Sch}/S)^{\mathrm{op}} \to \mathrm{Set} be a contravariant functor, the moduli functor. The set F(T)F(T) consists of the isomorphism classes of families of the objects to be classified, parametrized by the scheme TT. The functor FF acts on morphisms by pullback of families. For an SS-scheme MM, write hM=Hom⁡S(−,M)h_M = \Hom_S(-, M) for its functor of points.

  1. A fine moduli space for FF is an SS-scheme MM with an isomorphism of functors F≅hMF \cong h_M, that is, MM represents FF. The family 𝒰∈F(M)\mathcal{U} \in F(M) that corresponds to id⁡M\id_M is the universal family.
  2. A coarse moduli space for FF is an SS-scheme MM with a natural transformation Φ ⁣:F→hM\Phi \colon F \to h_M such that:
    1. for every algebraically closed field Ω\Omega and morphism Spec⁡Ω→S\operatorname{Spec} \Omega \to S, the map Φ ⁣:F(Spec⁡Ω)→M(Ω)\Phi \colon F(\operatorname{Spec} \Omega) \to M(\Omega) is a bijection;
    2. for every SS-scheme NN and natural transformation Ψ ⁣:F→hN\Psi \colon F \to h_N, there is a unique morphism f ⁣:M→Nf \colon M \to N with Ψ=hf∘Φ\Psi = h_f \circ \Phi.

Remarks

A fine moduli space is also a coarse one, and every family over TT is the pullback of 𝒰\mathcal{U} along a unique map T→MT \to M. Example: let F(T)F(T) be the set of line subbundles L⊂𝒪Tn+1L \subset \mathcal{O}_T^{n+1} with locally free quotient. Then ℙSn\mathbb{P}^n_S is a fine moduli space for FF, with universal family the tautological line bundle 𝒪(−1)⊂𝒪n+1\mathcal{O}(-1) \subset \mathcal{O}^{n+1}. Non-example: elliptic curves over an algebraically closed field of characteristic 00 have no fine moduli space. Every elliptic curve has the automorphism −1-1, and this gives nontrivial families whose fibers are all isomorphic. The affine line, through the j-invariant, is a coarse moduli space for them. Automorphisms are the usual obstruction to fine moduli spaces, and moduli stacks are a way around it.

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Functor, concept, no evidence levelFunctorconceptFunctor of points, concept, no evidence levelFunctor of pointsconceptGroup, concept, no evidence levelGroupconceptRepresentable functor, concept, no evidence levelRepresentable functorconceptScheme, concept, no evidence levelSchemeconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionField, definition, no evidence levelFielddefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionSplit polynomial, definition, no evidence levelSplit polynomialdefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionAlgebraically closed field, definition, no evidence levelAlgebraically closed …definitionModuli space, definition, no evidence levelModuli spacedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Functor concept
  2. Functor of points concept
  3. Group concept
  4. Representable functor concept
  5. Scheme concept
  6. Ring definition
  7. Commutative ring definition
  8. Polynomial ring definition
  9. Unit of a ring definition
  10. Zero divisor definition
  11. Field definition
  12. Polynomial evaluation definition
  13. Integral domain definition
  14. Root of a polynomial definition
  15. Irreducible element definition
  16. Split polynomial definition
  17. Irreducible polynomial definition
  18. Algebraically closed field definition
  19. Moduli space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Moduli_space
About this page
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Kinddefinition
WritingW0 stub
MSC14D22, 14D20
DefinesModuli space
UsesScheme, Functor, Functor of points, Representable functor, Algebraically closed field
Changed2 November 2026

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