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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.