Hilbert scheme theory

All labels observed (6)

Label	Occurrences
Hilbert schemes	2
Hilbert scheme of curves	1
Hilbert scheme of points	1
Hilbert scheme theory canonical	1
Hilbert schemes of points on surfaces	1
Quot schemes	1

How this entity was disambiguated

This entity first appeared as the object of triple T9297095 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Hilbert scheme theory
Context triple: [Castelnuovo–Mumford regularity, usedIn, Hilbert scheme theory]

A. Grothendieck’s scheme-theoretic framework
Grothendieck’s scheme-theoretic framework is a foundational reformulation of algebraic geometry that generalizes varieties using schemes, enabling powerful tools like sheaf theory, cohomology, and modern number-theoretic applications.
B. Foundations of Algebraic Geometry
Foundations of Algebraic Geometry is a landmark mathematical treatise that systematically developed the modern foundations of algebraic geometry and profoundly influenced the field’s subsequent evolution.
C. Chevalley’s theorem in algebraic geometry
Chevalley’s theorem in algebraic geometry is a fundamental result stating that the image of a morphism of finite type between schemes (or varieties) is a constructible set, playing a key role in understanding how geometric properties behave under mappings.
D. Topological Methods in Algebraic Geometry
Topological Methods in Algebraic Geometry is a foundational mathematical monograph by Friedrich Hirzebruch that applies topological techniques, particularly characteristic classes and cobordism theory, to problems in algebraic geometry.
E. Brill–Noether theory
Brill–Noether theory is a branch of algebraic geometry that studies linear series on algebraic curves, particularly the existence and dimension of spaces of special divisors and maps to projective spaces.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Hilbert scheme theory
Target entity description: 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.

A. Grothendieck’s scheme-theoretic framework
Grothendieck’s scheme-theoretic framework is a foundational reformulation of algebraic geometry that generalizes varieties using schemes, enabling powerful tools like sheaf theory, cohomology, and modern number-theoretic applications.
B. Foundations of Algebraic Geometry
Foundations of Algebraic Geometry is a landmark mathematical treatise that systematically developed the modern foundations of algebraic geometry and profoundly influenced the field’s subsequent evolution.
C. Chevalley’s theorem in algebraic geometry
Chevalley’s theorem in algebraic geometry is a fundamental result stating that the image of a morphism of finite type between schemes (or varieties) is a constructible set, playing a key role in understanding how geometric properties behave under mappings.
D. Topological Methods in Algebraic Geometry
Topological Methods in Algebraic Geometry is a foundational mathematical monograph by Friedrich Hirzebruch that applies topological techniques, particularly characteristic classes and cobordism theory, to problems in algebraic geometry.
E. Brill–Noether theory
Brill–Noether theory is a branch of algebraic geometry that studies linear series on algebraic curves, particularly the existence and dimension of spaces of special divisors and maps to projective spaces.
F. None of above. chosen

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 ⓘ
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 ⓘ
fieldOfStudy	Hilbert schemes ⓘ linked to: Hilbert scheme theory moduli of subschemes ⓘ
formalizedIn	Éléments de géométrie algébrique ⓘ
hasKeyObject	Hilbert scheme of curves ⓘ linked to: Hilbert scheme theory Hilbert scheme of points ⓘ linked to: Hilbert scheme theory Hilbert scheme of subschemes with fixed Hilbert polynomial ⓘ
historicalRoot	David Hilbert ⓘ
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 ⓘ Gromov–Witten theory ⓘ linked to: Gromov–Witten invariants 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 ⓘ Grassmannians ⓘ linked to: Grassmann manifolds Grothendieck’s representability theorem ⓘ Noetherian schemes ⓘ Quot schemes ⓘ linked to: Hilbert scheme theory coherent sheaves ⓘ flat families of schemes ⓘ functor of points ⓘ projective schemes ⓘ representable functors ⓘ

How these facts were elicited