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 (1)
| Label | Occurrences |
|---|---|
| Green’s conjecture canonical | 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.
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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.
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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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| provedBy | Claire Voisin NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| relatedTo |
Brill–Noether theory
NERFINISHED
ⓘ
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 GENERATED ⓘ |
| 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
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Instruction
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.