Hilbert scheme theory

E790521

branch of algebraic geometry

Hilbert scheme theory is a branch of algebraic geometry that studies parameter spaces representing families of subschemes of projective space, capturing how such geometric objects vary in moduli.

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Statements (49)

Predicate	Object
instanceOf	branch of algebraic geometry ⓘ
aimsAt	representing functors of families of subschemes ⓘ
appliesTo	closed subschemes of projective space ⓘ curves in projective space ⓘ higher-dimensional projective varieties ⓘ subschemes of projective space ⓘ surfaces in projective space ⓘ zero-dimensional subschemes ⓘ
assumes	base scheme is Noetherian ⓘ
basedOn	Hilbert polynomial NERFINISHED ⓘ
clarifies	structure of moduli spaces ⓘ variation of algebraic subschemes in families ⓘ
concerns	connected components of Hilbert schemes ⓘ existence of Hilbert schemes ⓘ properties of Hilbert schemes such as smoothness and irreducibility ⓘ
developedBy	Alexander Grothendieck NERFINISHED ⓘ
fieldOfStudy	Hilbert schemes NERFINISHED ⓘ moduli of subschemes ⓘ
formalizedIn	Éléments de géométrie algébrique ⓘ
hasKeyObject	Hilbert scheme of curves NERFINISHED ⓘ Hilbert scheme of points NERFINISHED ⓘ Hilbert scheme of subschemes with fixed Hilbert polynomial ⓘ
historicalRoot	David Hilbert NERFINISHED ⓘ
relatedTo	birational geometry ⓘ deformation theory ⓘ enumerative geometry ⓘ geometric invariant theory ⓘ intersection theory ⓘ moduli of curves ⓘ moduli of higher-dimensional varieties ⓘ
studies	families of subschemes of projective space ⓘ moduli problems in algebraic geometry ⓘ parameter spaces of subschemes ⓘ
usedIn	Donaldson–Thomas theory NERFINISHED ⓘ Gromov–Witten theory NERFINISHED ⓘ construction of moduli spaces of curves ⓘ construction of moduli spaces of sheaves ⓘ construction of moduli spaces of stable maps ⓘ string theory compactifications ⓘ
usesConcept	Castelnuovo–Mumford regularity NERFINISHED ⓘ Grassmannians NERFINISHED ⓘ Grothendieck’s representability theorem NERFINISHED ⓘ Noetherian schemes ⓘ Quot schemes NERFINISHED ⓘ coherent sheaves ⓘ flat families of schemes ⓘ functor of points ⓘ projective schemes ⓘ representable functors ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Castelnuovo–Mumford regularity → usedIn → Hilbert scheme theory ⓘ