Serre’s theorem on projective embeddings via ample line bundles

All labels observed (5)

Label	Occurrences
Serre’s cohomological criterion for ampleness	2
Kodaira embedding theorem	1
Serre’s theorem on projective embeddings via ample line bundles canonical	1
Serre’s theorem on projective normality via cohomology	1
Serre’s theorem on projective schemes and graded rings	1

How this entity was disambiguated

This entity first appeared as the object of triple T10732913 — 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: Serre’s theorem on projective embeddings via ample line bundles
Context triple: [GAGA (Géométrie Algébrique et Géométrie Analytique), relatedConcept, Serre’s theorem on projective embeddings via ample line bundles]

A. Lazarsfeld’s Positivity in Algebraic Geometry
Lazarsfeld’s *Positivity in Algebraic Geometry* is a two-volume monograph that serves as a standard modern reference on the theory of positivity for line bundles and divisors in algebraic geometry, integrating techniques from cohomology, vanishing theorems, and birational geometry.
B. Chevalley’s theorem in algebraic geometry
Chevalley’s theorem in algebraic geometry is a fundamental result stating that the image of a morphism of finite type between schemes (or varieties) is a constructible set, playing a key role in understanding how geometric properties behave under mappings.
C. Topological Methods in Algebraic Geometry
Topological Methods in Algebraic Geometry is a foundational mathematical monograph by Friedrich Hirzebruch that applies topological techniques, particularly characteristic classes and cobordism theory, to problems in algebraic geometry.
D. Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
E. Brill–Noether theory
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.
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: Serre’s theorem on projective embeddings via ample line bundles
Target entity description: Serre’s theorem on projective embeddings via ample line bundles is a foundational result in algebraic geometry that characterizes when a variety can be embedded into projective space using sufficiently high tensor powers of an ample line bundle.

A. Lazarsfeld’s Positivity in Algebraic Geometry
Lazarsfeld’s *Positivity in Algebraic Geometry* is a two-volume monograph that serves as a standard modern reference on the theory of positivity for line bundles and divisors in algebraic geometry, integrating techniques from cohomology, vanishing theorems, and birational geometry.
B. Chevalley’s theorem in algebraic geometry
Chevalley’s theorem in algebraic geometry is a fundamental result stating that the image of a morphism of finite type between schemes (or varieties) is a constructible set, playing a key role in understanding how geometric properties behave under mappings.
C. Topological Methods in Algebraic Geometry
Topological Methods in Algebraic Geometry is a foundational mathematical monograph by Friedrich Hirzebruch that applies topological techniques, particularly characteristic classes and cobordism theory, to problems in algebraic geometry.
D. Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
E. Brill–Noether theory
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.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	result in projective geometry ⓘ theorem in algebraic geometry ⓘ
appearsIn	Serre’s work on coherent algebraic sheaves ⓘ
assumes	Noetherian base ring ⓘ existence of an ample invertible sheaf ⓘ properness of the underlying scheme over the base ring ⓘ
characterizes	projective embeddings via high tensor powers of ample line bundles ⓘ when a scheme with an ample line bundle is projective ⓘ
coreStatement	for an ample line bundle L on a proper scheme X over a Noetherian ring, L^n is very ample for n sufficiently large ⓘ for n sufficiently large, higher cohomology groups of coherent sheaves twisted by L^n vanish ⓘ for n sufficiently large, the global sections of L^n give a closed immersion of X into projective space ⓘ sufficiently high tensor powers of an ample line bundle define a projective embedding ⓘ
field	algebraic geometry ⓘ projective algebraic geometry ⓘ
formalizedIn	EGA II by Grothendieck and Dieudonné ⓘ linked to: Éléments de géométrie algébrique
generalizes	classical results on embeddings of projective varieties ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	Serre vanishing theorem ⓘ existence of projective embeddings for varieties with ample line bundles ⓘ
involvesConcept	Noetherian scheme ⓘ linked to: Noetherian space Serre vanishing ⓘ linked to: Serre vanishing theorem Serre’s cohomological criterion for ampleness ⓘ linked to: Serre’s theorem on projective embeddings via ample line bundles ample line bundle ⓘ coherent sheaf ⓘ cohomology of coherent sheaves ⓘ global section of a line bundle ⓘ projective embedding ⓘ projective space ⓘ projective variety ⓘ quasi-coherent sheaf ⓘ scheme ⓘ tensor power of a line bundle ⓘ very ample line bundle ⓘ
namedAfter	Jean-Pierre Serre ⓘ
relatedTo	Castelnuovo–Mumford regularity ⓘ Kodaira embedding theorem ⓘ Nakai–Moishezon criterion ⓘ Serre’s GAGA theorem ⓘ linked to: Serre’s cohomological methods in algebraic geometry Serre’s theorem on affineness via global sections ⓘ
requiresTool	graded rings and Proj construction ⓘ sheaf cohomology ⓘ
usedFor	constructing projective models of varieties ⓘ defining projective morphisms via relatively ample line bundles ⓘ embedding schemes into projective space ⓘ proving projectivity criteria ⓘ showing that Proj of a graded ring is projective ⓘ

How these facts were elicited