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Hilbert Polynomials of Projective Schemes

We introduce localization and sheaves to define projective schemes, and in particular the projective n-space. Afterwards, we define closed subschemes of projective space and show that they arise from quotients of graded rings by homogeneous ideals. We then define the Hilbert function and Hilbert polynomial to determine several invariants of closed subschemes of projective space: their degree, dimension, and arithmetic genus. Finally, we provide numerous examples with explicit computations, finding the invariants of hypersurfaces, curves, the twisted cubic and more.

Identiferoai:union.ndltd.org:UPSALLA1/oai:DiVA.org:kth-315359
Date January 2022
CreatorsMa, Hemming
PublisherKTH, Fysik
Source SetsDiVA Archive at Upsalla University
LanguageEnglish
Detected LanguageEnglish
TypeStudent thesis, info:eu-repo/semantics/bachelorThesis, text
Formatapplication/pdf
Rightsinfo:eu-repo/semantics/openAccess
RelationTRITA-SCI-GRU ; 2022:111

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