Clifford’s theorem
E898501
Clifford’s theorem is a fundamental result in algebraic geometry that constrains the dimension of special linear series on algebraic curves in terms of their degree.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Clifford index of a curve | 1 |
| Clifford’s theorem canonical | 1 |
| Clifford’s theorem for line bundles | 1 |
| Clifford’s theorem for metric graphs | 1 |
| Clifford’s theorem in tropical geometry | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10992014 — 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: Clifford’s theorem Context triple: [Brill–Noether theory, usesConcept, Clifford’s theorem]
-
A.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
B.
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.
-
C.
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.
-
D.
Plücker formulas
Plücker formulas are classical algebraic geometry relations that connect the degree and singularities of plane algebraic curves with the invariants of their dual curves.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- 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: Clifford’s theorem Target entity description: Clifford’s theorem is a fundamental result in algebraic geometry that constrains the dimension of special linear series on algebraic curves in terms of their degree.
-
A.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
B.
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.
-
C.
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.
-
D.
Plücker formulas
Plücker formulas are classical algebraic geometry relations that connect the degree and singularities of plane algebraic curves with the invariants of their dual curves.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
result in the theory of algebraic curves
ⓘ
theorem ⓘ theorem in algebraic geometry ⓘ |
| appliesTo |
divisors on smooth projective curves
ⓘ
smooth projective algebraic curves ⓘ |
| assumes | algebraically closed base field (in standard formulations) ⓘ |
| characterizesEqualityCase |
equality holds only for hyperelliptic curves or trivial cases
ⓘ
if equality holds for a special divisor on a nontrivial curve, then the curve is hyperelliptic ⓘ |
| codomainObject | space of global sections of a line bundle ⓘ |
| concerns |
Clifford index of a curve
ⓘ
linked to:
Clifford’s theorem
dimension of complete linear systems ⓘ divisors on algebraic curves ⓘ special linear series on algebraic curves ⓘ |
| domainObject | smooth projective curve over an algebraically closed field ⓘ |
| field |
algebraic curves
ⓘ
algebraic geometry ⓘ |
| givesInequality |
h^0(C, O_C(D)) ≤ 1 + deg(D)/2 for special divisors D on a curve C
ⓘ
l(D) ≤ 1 + deg(D)/2 for special divisors D ⓘ |
| hasVariant |
Clifford’s theorem for line bundles
ⓘ
linked to:
Clifford’s theorem
Clifford’s theorem for metric graphs ⓘ
linked to:
Clifford’s theorem
Clifford’s theorem in tropical geometry ⓘ
linked to:
Clifford’s theorem
|
| historicalPeriod | 19th century mathematics ⓘ |
| implies |
constraints on existence of low-degree maps to projective spaces
ⓘ
upper bound on the dimension of special linear series ⓘ |
| mathematicalSubjectClassification |
14H51
ⓘ
14H55 ⓘ |
| namedAfter | William Kingdon Clifford ⓘ |
| relatedConcept |
gonality of a curve
ⓘ
special linear series g^r_d ⓘ |
| relatedTheorem |
Brill–Noether theorem
ⓘ
linked to:
Brill–Noether theory
Noether’s theorem on canonical curves ⓘ Riemann–Roch theorem ⓘ |
| relatesConcept |
Brill–Noether theory
ⓘ
Riemann–Roch theorem ⓘ canonical divisor ⓘ degree of a divisor ⓘ dimension of a linear series ⓘ genus of a curve ⓘ nonspecial divisors ⓘ special divisors ⓘ |
| standardReference |
Arbarello–Cornalba–Griffiths–Harris, Geometry of Algebraic Curves
ⓘ
Robin Hartshorne, Algebraic Geometry ⓘ |
| strengthens | information obtained from the Riemann–Roch theorem for special divisors ⓘ |
| usedIn |
Brill–Noether theory of linear series
ⓘ
linked to:
Brill–Noether theory
classification of algebraic curves ⓘ definition and study of the Clifford index ⓘ proofs of results about gonality of curves ⓘ study of hyperelliptic curves ⓘ |
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: Clifford’s theorem Description of subject: Clifford’s theorem is a fundamental result in algebraic geometry that constrains the dimension of special linear series on algebraic curves in terms of their degree.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Clifford’s theorem
linked to: Clifford’s theorem
linked to: Clifford’s theorem
linked to: Clifford’s theorem