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.

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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

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 ⓘ

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