Clifford’s theorem

E898501

result in the theory of algebraic curves theorem theorem in algebraic geometry

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.

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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 ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Brill–Noether theory → usesConcept → Clifford’s theorem ⓘ