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

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 ⓘ

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

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 AF+BG theorem in plane curves