Hilbert scheme theory
E790521
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hilbert scheme theory canonical | 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.
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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.
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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.
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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 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hilbert scheme theory Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.