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

Link

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

Cited from

mathematics

Cited by

10 citations across 10 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

• 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.
• 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.
• Regular scheme
  • A regular scheme is nonsingular in the intrinsic local-ring sense at every point.
• 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.
• 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.

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