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 (1)
| Label | Occurrences |
|---|---|
| Clifford’s theorem canonical | 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]
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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.
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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.
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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.
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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
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
Clifford’s theorem for metric graphs NERFINISHED ⓘ Clifford’s theorem in tropical geometry NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| relatedConcept |
gonality of a curve
ⓘ
special linear series g^r_d ⓘ |
| relatedTheorem |
Brill–Noether theorem
NERFINISHED
ⓘ
Noether’s theorem on canonical curves NERFINISHED ⓘ Riemann–Roch theorem NERFINISHED ⓘ |
| relatesConcept |
Brill–Noether theory
NERFINISHED
ⓘ
Riemann–Roch theorem NERFINISHED ⓘ 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
NERFINISHED
ⓘ
Robin Hartshorne, Algebraic Geometry ⓘ |
| strengthens | information obtained from the Riemann–Roch theorem for special divisors ⓘ |
| usedIn |
Brill–Noether theory of linear series
NERFINISHED
ⓘ
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
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Instruction
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.