Serre’s theorem on projective embeddings via ample line bundles

E883482

result in projective geometry theorem in algebraic geometry

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.

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All labels observed (2)

Label	Occurrences
Kodaira embedding theorem	1
Serre’s theorem on projective embeddings via ample line bundles canonical	1

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 NERFINISHED ⓘ
generalizes	classical results on embeddings of projective varieties ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	Serre vanishing theorem NERFINISHED ⓘ existence of projective embeddings for varieties with ample line bundles ⓘ
involvesConcept	Noetherian scheme NERFINISHED ⓘ Serre vanishing NERFINISHED ⓘ Serre’s cohomological criterion for ampleness NERFINISHED ⓘ 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 NERFINISHED ⓘ
relatedTo	Castelnuovo–Mumford regularity NERFINISHED ⓘ Kodaira embedding theorem NERFINISHED ⓘ Nakai–Moishezon criterion NERFINISHED ⓘ Serre’s GAGA theorem NERFINISHED ⓘ Serre’s theorem on affineness via global sections NERFINISHED ⓘ
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 ⓘ

Referenced by (2)

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

GAGA (Géométrie Algébrique et Géométrie Analytique) → relatedConcept → Serre’s theorem on projective embeddings via ample line bundles ⓘ

Kunihiko Kodaira → notableWork → Serre’s theorem on projective embeddings via ample line bundles ⓘ

this entity surface form: Kodaira embedding theorem