Algebraic Geometry

Hartshorne, R. (1977). Algebraic Geometry. Springer.

Type

Book

Intellectual base

Review or monograph

Year

1977

DOI

10.1007/978-1-4757-3849-0

ISBN

9781441928078

DOI registrant

Springer Nature

Link

https://doi.org/10.1007/978-1-4757-3849-0

Cited from

mathematics

Cited by

19 citations across 19 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Local-to-Global Aggregation
  • Sheaf theory / glueing — local sections agreeing on overlaps glue uniquely to a global section; agreement-on-overlaps is the explicit discipline.
• Neighborhood
  • The intervention this licenses is to attack any complex global object by carving it into overlapping local neighborhoods, solving locally, and checking the glue — the strategy behind sheaves, atlases, and finite-element meshes alike.

    This sourcePresents sheaves and the systematic gluing of local data into global objects — the attack-by-local-neighborhoods-then-check-the-glue strategy behind atlases and sheaves.

Domain-specific

• Algebraic Variety
  • In its classical affine form, it begins as the common zero set of polynomials in an affine space.

    This sourceDevelops affine and projective varieties, coordinate rings, regular maps, dimension, and gluing.

• Complete variety
  • Complete variety is an algebraic variety whose projection after product with any variety is a closed map, the algebro-geometric analogue of compactness.
• Degeneration (algebraic geometry)
  • Degeneration turns a difficult object into a limit within an algebraic family so geometry can be studied through the special fiber, while distinguishing genuine flat specialization from arbitrary collapse.
• Euler sequence
  • The conclusion depends on the exact sequence, not merely on dimension counting.
• Fiber Product of Schemes

  • This sourceGraduate Texts in Mathematics 52. Springer, 1977. See Chapter II, especially Theorem II.3.3 and the discussion of fibers and base change.

• Ideal sheaf
  • … \(\operatorname{Spec}(A/I)\). to If two closed subschemes have ideal sheaves \(\mathcal I\) and \(\mathcal J\), their scheme-theoretic intersection is defined by \(\mathcal I+\mathcal J\), while the product or intersection of ideals records different union or thickening behavior. demonstrates that continuity.
• Morphism of finite type
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Projective line
  • … module, equivalence relation on nonzero pairs, homogeneous-coordinate convention, affine charts and transition, point or points at infinity, incidence, and automorphism group are explicit. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Projectivization
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Pseudo-canonical variety
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Regular scheme
  • A regular scheme is nonsingular in the intrinsic local-ring sense at every point.
• Ring
  • Prime and maximal ideals within a commutative ring generalize the notion of prime and irreducible elements and are the objects through which algebraic geometry passes: the spectrum of a commutative ring (the set of prime ideals with the Zariski topology) is the foundational construction of scheme theory

    This sourceThe spectrum of a commutative ring as the affine scheme from which scheme theory is built (Ch. II).

    Supported in partVerified against the source

    Read by an automated reader; not a human review. How support was checked

• Ringed Space
  • These are literal applications of the same domain machinery, not cross-domain metaphors.
• Ruled Surface
  • In algebraic geometry, “ruled surface” has related projective/birational formulations; those conventions must not be silently interchanged with the smooth embedded definition.
• Sheaf of algebras
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Specialization preorder
  • The specialization preorder converts topological indistinguishability and closure into order: it becomes antisymmetric exactly for T0 spaces and collapses to equality for T1 spaces.
• Topological Space
  • . Algebraic geometry — the Zariski topology, whose closed sets are polynomial zero-loci, brings the topological machinery to varieties where no metric is available

    This sourceThe Zariski topology, whose closed sets are the zero loci of polynomials, as the topology of algebraic varieties, and its failure to be Hausdorff.

    SupportedVerified against the source

    Read by an automated reader; not a human review. How support was checked

Verification

Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.

Does it back the claim? Read against the text for 2 of 19 citations: 1 supported, 1 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

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Links previously used in the corpus

Before the registry existed this work was also linked 1 other way.

• https://link.springer.com/book/10.1007/978-1-4757-3849-0 ×1

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