Quasi-projective variety¶
A quasi-projective variety is a locally closed subvariety of projective space, equivalently an intersection of a Zariski-open and Zariski-closed subset.
Core Idea¶
A quasi-projective variety is an algebraic variety that can be realized as a locally closed subset of a projective variety. Concretely, inside some projective space it has the form \(U\cap Z\), where \(U\) is Zariski open and \(Z\) is Zariski closed. Equivalently, it is open in its projective closure. The analogous scheme-theoretic definition uses a locally closed subscheme of projective space. This condition places quasi-projective varieties between the affine and projective cases without reducing them to either. The embedding condition is constitutive.
Scope of Application¶
Quasi-projective variety applies to algebraic varieties admitting a locally closed immersion into projective space, equivalently an open realization inside a projective closure; local affineness alone and informal resemblance to a projective object do not establish the property. - Projective varieties. Every projective variety is quasi-projective by taking the relevant open part to be its entire projective image. - Affine varieties. An affine variety becomes a locally closed projective subvariety through the standard affine open of projective space and its projective completion. - Open parts of projective varieties. Deleting a Zariski-closed boundary from a projective variety preserves quasi-projectivity even when projectivity is lost. - Intermediate nonaffine, nonprojective varieties. The complement of a point in projective space of dimension at least two exemplifies a genuinely quasi-projective habitat outside both narrower classes.
Clarity¶
Naming a quasi-projective variety identifies an embedding property, not a vague degree of resemblance to a projective variety. The variety must occur as a locally closed subvariety of projective space—equivalently, as the intersection of a Zariski-open subset with a Zariski-closed subset, or as an open subvariety of its projective closure.
Manages Complexity¶
Algebraic varieties may be presented by many local affine charts, equations, deletions, and embeddings. Quasi-projectivity compresses that presentation problem into a locally closed immersion: realize the variety as U ∩ Z, with U Zariski open and Z Zariski closed in projective space, or equivalently as an open part of its projective closure. This pair records both the ambient compactifying geometry and the boundary removed from it.
Abstract Reasoning¶
A locally closed presentation supports a direct classification inference: from an exhibited variety X = U ∩ Z, with U Zariski open and Z Zariski closed in projective space, infer that X is quasi-projective. Conversely, a claimed quasi-projective variety directs the geometer to seek such an immersion, or equivalently to realize it as an open subvariety of a projective closure. The concept also organizes intervention and boundary reasoning.
Knowledge Transfer¶
Within algebraic geometry, quasi-projectivity transfers literally across varieties, base presentations, open deletions, projective closures, and the analogous scheme-theoretic setting. The same certificate carries: exhibit a locally closed immersion into projective space, write the object as U ∩ Z, or realize it as open in a projective closure. Beyond this named class, the defensible reach is (B) a shared abstract mechanism under pattern: an object can inherit useful ambient structure by being represented as the intersection of an open and a closed part.
Relationships to Other Abstractions¶
Current abstraction Quasi-projective variety Domain-specific
Parents (1) — more general patterns this builds on
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Quasi-projective variety is a kind of Pattern Prime
The carriers are algebraic varieties or schemes considered at the granularity of an immersion into projective space.
Hierarchy path (1) — routes to 1 parentless root
- Quasi-projective variety → Pattern → Abstraction
Neighborhood in Abstraction Space¶
Quasi-projective variety sits in a sparse region of the domain-specific corpus (66th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Algebraic Varieties & Arithmetic Cohomology (7 abstractions)
Nearest neighbors
- Secant Variety — 0.85
- Closed Immersion — 0.85
- Contraction Morphism — 0.85
- Mordellic Variety — 0.84
- Complete variety — 0.84
Computed from structural-signature embeddings · 2026-10-08