Brill–Noether theory

E259773

branch of algebraic geometry

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (4)

Label	Occurrences
Brill–Noether loci	1
Brill–Noether number	1
Brill–Noether theory canonical	1
Green’s conjecture on syzygies	1

How this entity was disambiguated

This entity first appeared as the object of triple T2364573 — 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: Brill–Noether theory
Context triple: [Riemann–Roch theorem, usedIn, Brill–Noether theory]

A. Foundations of Algebraic Geometry
Foundations of Algebraic Geometry is a landmark mathematical treatise that systematically developed the modern foundations of algebraic geometry and profoundly influenced the field’s subsequent evolution.
B. 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.
C. Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
D. Castelnuovo–Mumford regularity
Castelnuovo–Mumford regularity is an invariant in commutative algebra and algebraic geometry that measures the complexity of the minimal graded free resolution of a module or sheaf, often used to control vanishing of cohomology and bounds on generators.
E. Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
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: Brill–Noether theory
Target entity description: 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.

A. Foundations of Algebraic Geometry
Foundations of Algebraic Geometry is a landmark mathematical treatise that systematically developed the modern foundations of algebraic geometry and profoundly influenced the field’s subsequent evolution.
B. 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.
C. Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
D. Castelnuovo–Mumford regularity
Castelnuovo–Mumford regularity is an invariant in commutative algebra and algebraic geometry that measures the complexity of the minimal graded free resolution of a module or sheaf, often used to control vanishing of cohomology and bounds on generators.
E. Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
F. None of above. chosen

Statements (50)

Predicate	Object
instanceOf	branch of algebraic geometry ⓘ
appliesTo	complex algebraic curves ⓘ smooth projective algebraic curves ⓘ
centralInvariant	Brill–Noether theory self-linksurface differs ⓘ surface form: Brill–Noether number
concerns	actual dimension of linear series spaces ⓘ existence of maps of given degree and dimension ⓘ expected dimension of linear series spaces ⓘ
developedBy	Alexander Brill ⓘ Max Noether ⓘ
field	algebraic geometry ⓘ
focusesOn	dimension of spaces of linear series ⓘ existence of linear series ⓘ moduli of linear series ⓘ spaces of special divisors ⓘ
furtherDevelopedBy	David Mumford ⓘ Enrico Arbarello ⓘ Joe Harris ⓘ Maurizio Cornalba ⓘ Phillip Griffiths ⓘ surface form: P. A. Griffiths Phillip Griffiths ⓘ
hasApplicationIn	classification of algebraic curves ⓘ construction of maps to projective spaces ⓘ
historicalDevelopment	late 19th century ⓘ
namedAfter	Alexander Brill ⓘ Max Noether ⓘ
relatedTo	Green’s conjecture ⓘ Petri’s theorem ⓘ Riemann surfaces ⓘ surface form: Riemann surface theory moduli theory ⓘ projective embeddings of curves ⓘ syzygies of curves ⓘ
studies	Brill–Noether theory self-linksurface differs ⓘ surface form: Brill–Noether loci linear series on algebraic curves ⓘ maps from algebraic curves to projective spaces ⓘ special divisors on algebraic curves ⓘ varieties of linear series ⓘ varieties of special divisors ⓘ
usesConcept	Clifford’s theorem ⓘ Hurwitz space ⓘ Jacobian varieties ⓘ surface form: Jacobian of a curve Jacobian varieties ⓘ surface form: Picard variety Riemann–Roch theorem ⓘ Weierstrass points ⓘ complete linear series ⓘ divisor on a curve ⓘ gonality of a curve ⓘ incomplete linear series ⓘ line bundle on a curve ⓘ moduli space of curves ⓘ special divisor ⓘ

How these facts were elicited

Referenced by (4)

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

Riemann–Roch theorem → usedIn → Brill–Noether theory ⓘ

Castelnuovo–Mumford regularity → relatedTo → Brill–Noether theory ⓘ

this entity surface form: Green’s conjecture on syzygies

Brill–Noether theory → centralInvariant → Brill–Noether theory self-linksurface differs ⓘ

this entity surface form: Brill–Noether number

Brill–Noether theory → studies → Brill–Noether theory self-linksurface differs ⓘ

this entity surface form: Brill–Noether loci

All labels observed (4)

How this entity was disambiguated Show

Statements (50)

How these facts were elicited Show

Referenced by (4)

How this entity was disambiguated

How these facts were elicited