Math 2.0

Algebraic geometry Cycles and intersection theory Object

Hilbert scheme

Builds on Moduli spaceProjective varietyProjective spaceFlat morphismProper morphismFunctor of pointsRepresentable functorHilbert polynomial

Description

Let kk be a field and let X⊂ℙknX \subset \mathbb{P}^n_k be a closed subscheme of projective space, for example a projective variety. The Hilbert functor of XX sends a kk-scheme TT to the set of closed subschemes Z⊂X×kTZ \subset X \times_k T such that the projection Z→TZ \to T is flat and proper. Grothendieck proved that this functor is representable. So there are a kk-scheme Hilb⁡(X)\operatorname{Hilb}(X) and a universal flat family 𝒵⊂X×Hilb⁡(X)\mathcal{Z} \subset X \times \operatorname{Hilb}(X). Every family Z⊂X×TZ \subset X \times T as above is the pullback of 𝒵\mathcal{Z} along a unique morphism T→Hilb⁡(X)T \to \operatorname{Hilb}(X). This is the Hilbert scheme of XX. It is a disjoint union Hilb⁡(X)=∐PHilb⁡P(X)\operatorname{Hilb}(X) = \coprod_P \operatorname{Hilb}^P(X) over the Hilbert polynomials PP of the fibers, and each Hilb⁡P(X)\operatorname{Hilb}^P(X) is a projective scheme. So the Hilbert scheme is a fine moduli space for closed subschemes of XX, in the sense of the functor of points.

Remarks

The hypersurfaces of degree dd in ℙn\mathbb{P}^n are the zero sets of nonzero homogeneous polynomials of degree dd up to scalars. Their Hilbert scheme is the projective space ℙN\mathbb{P}^N with N=(n+dd)−1N = \binom{n+d}{d} - 1, whose points are these polynomials up to scaling. The Hilbert scheme also exists for quasi-projective XX, with quasi-projective components. The Hilbert scheme of mm points in the plane, Hilb⁡m(𝔸2)\operatorname{Hilb}^m(\mathbb{A}^2), is smooth and irreducible of dimension 2m2m by a theorem of Fogarty. It resolves the singularities of the symmetric product of mm copies of 𝔸2\mathbb{A}^2.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

Show the path as a graph
Field, concept, no evidence levelFieldconceptFinitely generated module, concept, no evidence levelFinitely generated mo…conceptFlat morphism, concept, no evidence levelFlat morphismconceptFunctor of points, concept, no evidence levelFunctor of pointsconceptGraded ring, concept, no evidence levelGraded ringconceptHilbert polynomial, concept, no evidence levelHilbert polynomialconceptModuli space, concept, no evidence levelModuli spaceconceptPolynomials in several variables, concept, no evidence levelPolynomials in severa…conceptProjective space, concept, no evidence levelProjective spaceconceptProjective variety, concept, no evidence levelProjective varietyconceptProper morphism, concept, no evidence levelProper morphismconceptRepresentable functor, concept, no evidence levelRepresentable functorconceptHilbert–Serre theorem, statement, no evidence levelHilbert–Serre theoremstatementHilbert polynomial, definition, no evidence levelHilbert polynomialdefinitionHilbert scheme, object, no evidence levelHilbert schemeobject
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.
  • no evidence level
  1. Field concept
  2. Finitely generated module concept
  3. Flat morphism concept
  4. Functor of points concept
  5. Graded ring concept
  6. Hilbert polynomial concept
  7. Moduli space concept
  8. Polynomials in several variables concept
  9. Projective space concept
  10. Projective variety concept
  11. Proper morphism concept
  12. Representable functor concept
  13. Hilbert–Serre theorem statement
  14. Hilbert polynomial definition
  15. Hilbert scheme object

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert_scheme
About this page
TagXK866CAZHE
URIm2://XK866CAZHE
Kindobject
WritingW0 stub
MSC14C05, 14D22
Instance ofModuli space
UsesModuli space, Projective variety, Projective space, Flat morphism, Proper morphism, Functor of points, Representable functor, Hilbert polynomial
Same as
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.