Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem)

E186169

mathematical theorem result in algebraic geometry

Noether’s AF+BG theorem is a foundational result in algebraic geometry that provides conditions under which a polynomial vanishing on the intersection of two plane curves can be expressed as a linear combination of their defining equations.

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All labels observed (7)

Label	Occurrences
Noether’s AF+BG theorem in plane curves	1
Noether’s theorem in algebraic geometry	1
Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem) canonical	1
Noether’s theorem on canonical curves	1
Noether’s theorem on rationality of surfaces	1
Noether’s theorem on the canonical embedding of algebraic curves	1
Noether’s theorem relating differentials and divisors on curves	1

How this entity was disambiguated

This entity first appeared as the object of triple T1645083 — 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: Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem)
Context triple: [Max Noether, notableWork, Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem)]

A. Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
B. Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
C. Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
D. Noether's theorem
Noether's theorem is a fundamental result in theoretical physics and mathematics that links continuous symmetries of a physical system to corresponding conservation laws, such as energy or momentum conservation.
E. Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
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: Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem)
Target entity description: Noether’s AF+BG theorem is a foundational result in algebraic geometry that provides conditions under which a polynomial vanishing on the intersection of two plane curves can be expressed as a linear combination of their defining equations.

A. Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
B. Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
C. Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
D. Noether's theorem
Noether's theorem is a fundamental result in theoretical physics and mathematics that links continuous symmetries of a physical system to corresponding conservation laws, such as energy or momentum conservation.
E. Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
F. None of above. chosen

Statements (44)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in algebraic geometry ⓘ
alsoKnownAs	Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem) ⓘ surface form: Noether’s AF+BG theorem in plane curves Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem) ⓘ surface form: Noether’s theorem in algebraic geometry
appearsIn	classical treatments of plane algebraic curves ⓘ texts on birational geometry of surfaces ⓘ
assumes	control of intersection multiplicities at base points ⓘ finite intersection of the two curves ⓘ
concerns	divisors on plane curves ⓘ ideals in polynomial rings ⓘ intersections of plane curves ⓘ plane algebraic curves ⓘ polynomials in two variables ⓘ
domain	classical algebraic geometry ⓘ commutative algebra ⓘ
field	algebraic geometry ⓘ
givesConditionFor	expressing a polynomial as AF+BG ⓘ membership in the ideal generated by two polynomials ⓘ
historicalPeriod	early 20th century mathematics ⓘ
implies	a criterion for ideal membership in k[x,y] ⓘ relations between vanishing conditions and ideal generators ⓘ
influenceOn	development of modern algebraic geometry ⓘ study of plane Cremona transformations ⓘ theory of linear systems on surfaces ⓘ
involves	Bézout’s theorem ⓘ homogeneous polynomials ⓘ multiplicity of intersection ⓘ projective plane curves ⓘ
mathematicianAssociated	Emmy Noether ⓘ
namedAfter	Emmy Noether ⓘ
relatedTo	Cremona group of the projective plane ⓘ Hilbert’s Nullstellensatz ⓘ Noether normalization lemma ⓘ surface form: Noether’s normalization lemma syzygies of plane curves ⓘ
statementAbout	linear combinations of defining equations of curves ⓘ polynomials vanishing on the intersection of two plane curves ⓘ
typicalSetting	algebraically closed base field ⓘ polynomial ring k[x,y,z] with homogeneous coordinates ⓘ
usedFor	classifying plane algebraic curves ⓘ computing linear equivalence of divisors on plane curves ⓘ proving properties of plane Cremona transformations ⓘ resolving base points by blowing up ⓘ studying base points of linear systems ⓘ studying birational maps of the projective plane ⓘ

How these facts were elicited

Referenced by (7)

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

Max Noether → notableWork → Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem) ⓘ

this entity surface form: Noether’s theorem on canonical curves

Max Noether → notableWork → Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem) ⓘ

this entity surface form: Noether’s theorem on rationality of surfaces

Max Noether → notableFor → Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem) ⓘ

this entity surface form: Noether’s theorem relating differentials and divisors on curves

Max Noether → notableFor → Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem) ⓘ

this entity surface form: Noether’s theorem on the canonical embedding of algebraic curves

Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem) → alsoKnownAs → Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem) ⓘ

subject surface form: Noether’s AF+BG theorem

this entity surface form: Noether’s theorem in algebraic geometry