Green’s conjecture
E898503
Green’s conjecture is a central statement in algebraic geometry that predicts a precise relationship between the syzygies of the canonical embedding of a smooth projective curve and its Clifford index.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Green’s conjecture canonical | 1 |
| Green’s conjecture on syzygies of canonical curves | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10992028 — 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: Green’s conjecture Context triple: [Brill–Noether theory, relatedTo, Green’s conjecture]
-
A.
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.
-
B.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
-
C.
Lazarsfeld’s Positivity in Algebraic Geometry
Lazarsfeld’s *Positivity in Algebraic Geometry* is a two-volume monograph that serves as a standard modern reference on the theory of positivity for line bundles and divisors in algebraic geometry, integrating techniques from cohomology, vanishing theorems, and birational geometry.
-
D.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
E.
Serre’s theorem on projective embeddings via ample line bundles
Serre’s theorem on projective embeddings via ample line bundles is a foundational result in algebraic geometry that characterizes when a variety can be embedded into projective space using sufficiently high tensor powers of an ample line bundle.
- 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: Green’s conjecture Target entity description: Green’s conjecture is a central statement in algebraic geometry that predicts a precise relationship between the syzygies of the canonical embedding of a smooth projective curve and its Clifford index.
-
A.
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.
-
B.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
-
C.
Lazarsfeld’s Positivity in Algebraic Geometry
Lazarsfeld’s *Positivity in Algebraic Geometry* is a two-volume monograph that serves as a standard modern reference on the theory of positivity for line bundles and divisors in algebraic geometry, integrating techniques from cohomology, vanishing theorems, and birational geometry.
-
D.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
E.
Serre’s theorem on projective embeddings via ample line bundles
Serre’s theorem on projective embeddings via ample line bundles is a foundational result in algebraic geometry that characterizes when a variety can be embedded into projective space using sufficiently high tensor powers of an ample line bundle.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
conjecture in algebraic geometry
ⓘ
mathematical conjecture ⓘ |
| appliesTo | smooth projective curves over an algebraically closed field of characteristic zero ⓘ |
| concerns |
Clifford index of algebraic curves
ⓘ
syzygies of canonical embeddings of smooth projective curves ⓘ |
| context | canonical embedding of a smooth projective curve ⓘ |
| failsFor | hyperelliptic curves in its naive form ⓘ |
| field | algebraic geometry ⓘ |
| formulationUses |
Koszul cohomology
ⓘ
minimal graded free resolutions ⓘ |
| implies |
constraints on the Betti table of the canonical curve
ⓘ
sharp bounds on the first non-vanishing syzygies ⓘ |
| influenced | subsequent work on syzygies of curves and higher-dimensional varieties ⓘ |
| involvesInvariant |
Clifford index
ⓘ
gonality of curves ⓘ |
| namedAfter | Mark Green ⓘ |
| oftenStudiedOver | complex numbers ⓘ |
| openProblems |
behavior in positive characteristic
ⓘ
validity for special curves with low Clifford index ⓘ |
| predicts |
precise relationship between syzygies of canonical curves and their Clifford index
ⓘ
vanishing pattern of Koszul cohomology groups of canonical curves ⓘ |
| proposedBy | Mark Green ⓘ |
| provedBy | Claire Voisin ⓘ |
| provedFor |
general curve of given genus in characteristic zero
ⓘ
general curves of even genus ⓘ general curves of odd genus ⓘ |
| relatedConjecture | Green–Lazarsfeld conjecture on syzygies of line bundles ⓘ |
| relatedTo |
Brill–Noether theory
ⓘ
canonical ring of a curve ⓘ syzygies of projective varieties ⓘ |
| relates |
Clifford index of a smooth projective curve
ⓘ
minimal free resolution of the canonical ring ⓘ |
| requires | non-trivial Clifford index ⓘ |
| standardReference | Mark Green’s 1984 paper on Koszul cohomology and the geometry of projective varieties ⓘ |
| status | partially proved ⓘ |
| subfield |
canonical curves
ⓘ
syzygies of algebraic varieties ⓘ theory of algebraic curves ⓘ |
| typicalAssumption | curve is non-hyperelliptic ⓘ |
| typicalGenusCondition | genus at least 2 ⓘ |
| usesConcept |
canonical linear system
ⓘ
graded Betti numbers ⓘ minimal graded free resolution over a polynomial ring ⓘ |
| VoisinProofMethod | use of K3 surfaces and their hyperplane sections ⓘ |
| VoisinProofYear |
2002
ⓘ
2005 ⓘ |
| yearProposed | 1984 ⓘ |
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: Green’s conjecture Description of subject: Green’s conjecture is a central statement in algebraic geometry that predicts a precise relationship between the syzygies of the canonical embedding of a smooth projective curve and its Clifford index.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Green’s conjecture