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.

Version: v1 · 2026-09-28 · History
Domain-specific #: 7739
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Algebraic Geometry → Mathematics

Core Idea¶

A quasi-projective variety is an algebraic variety that can be realized as a locally closed subset of a projective variety.^[1] Concretely, inside some projective space it has the form \(U\cap Z\), where \(U\) is Zariski open and \(Z\) is Zariski closed.^[2] Equivalently, it is open in its projective closure.^[3] The analogous scheme-theoretic definition uses a locally closed subscheme of projective space.^[4]

This condition places quasi-projective varieties between the affine and projective cases without reducing them to either. Every affine variety is quasi-projective because affine space is a Zariski-open part of projective space, and every projective variety is quasi-projective by taking the open part to be the whole ambient space.^[5] Yet a quasi-projective variety need be neither affine nor projective; deleting a point from projective space of dimension at least two gives a standard example.^[6]

The embedding condition is constitutive. Being locally affine is necessary but not by itself the stated definition, since that local property is shared much more broadly. Nor does “quasi-projective” mean approximately projective or merely similar to a projective variety. Recognition requires an actual locally closed immersion into a projective space or projective variety, preserving both the Zariski-open and Zariski-closed roles in the construction.

Structural Signature¶

Sig role-phrases:

candidate variety — the algebraic variety or scheme whose quasi-projective status is to be established.
projective ambient space — the projective variety or projective space into which the candidate is placed.
locally closed immersion — the embedding that identifies the candidate with a locally closed subobject of that ambient space.
Zariski-open part — the open subset U contributing the retained locus of the presentation.
Zariski-closed part — the closed subset Z supplying the algebraic ambient equations.
intersection certificate — the equality X = U ∩ Z that directly witnesses quasi-projectivity.
projective closure — a projective completion in which the candidate can equivalently occur as an open subvariety.
projective branch — the special case in which the open part is the whole relevant ambient locus.
affine branch — the special case embedded through the standard affine open of projective space.
intermediate branch — a locally closed variety that is neither affine nor projective.
scheme-theoretic branch — the analogous locally closed subscheme formulation with its morphism structure retained.
classification guarantee — exhibiting a valid locally closed projective immersion is sufficient to establish quasi-projectivity.
local-affineness boundary — affine neighborhoods at every point do not alone supply the required global embedding.
presentation limitation — quasi-projectivity guarantees neither a canonical immersion nor invariance of the visible complement across different projective presentations.

What It Is Not¶

Not “approximately projective.” The prefix marks an exact embedding property: the variety must be a locally closed subvariety of projective space or, equivalently, open in a projective closure.
Not necessarily projective. The Zariski-open part can be proper, so deleting a boundary from a projective variety may preserve quasi-projectivity while destroying projectivity.
Not necessarily affine. Affine varieties form one branch of the class, but locally closed projective presentations also include varieties that are neither affine nor projective.
Not established by local affineness alone. Having affine neighborhoods is a broader local property and does not by itself exhibit the required global locally closed immersion.
Not an arbitrary subset of projective space. The image must be locally closed—expressible as (U\cap Z) with (U) Zariski open and (Z) Zariski closed—and must retain the relevant algebraic variety or scheme structure.
Not tied to one canonical embedding. Different projective immersions or closures may display different equations and boundary complements while witnessing the same quasi-projective property.
Not a claim that the visible complement is intrinsic. The removed boundary belongs to a chosen projective presentation; quasi-projectivity guarantees the existence of a presentation, not invariance of all its presentation data.

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.
Principal affine opens — complements such as Aⁿ \ {f = 0} can be realized as affine hypersurfaces by adjoining a coordinate satisfying xₙ₊₁f = 1, supplying explicit quasi-projective presentations.
Quasi-affine varieties — varieties isomorphic to open subsets of affine varieties occupy an important subfamily within the quasi-projective class.
Scheme theory — the analogous notion applies to schemes that occur as locally closed subschemes of projective space, with the scheme structure and immersion retained.
Projective-closure methods — a chosen closure and its removed boundary let projective ambient techniques be applied to an open variety, while different closures or embeddings need not produce the same visible complement.

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. Merely being locally affine does not supply that embedding, because the local property is shared by a wider class of varieties.

The term lets an algebraic geometer ask: What locally closed immersion realizes this variety inside projective space? Exhibiting the open and closed pieces answers the classification directly and shows why every affine and every projective variety qualifies. It also makes the intermediate cases legible: deleting a point from projective space of dimension at least two can preserve quasi-projectivity while yielding a variety that is neither affine nor projective.

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.

The condition makes the main inclusion regimes readable. Projective varieties use the whole closed ambient object; affine varieties enter through the standard affine open in projective space; genuinely intermediate examples remain locally closed while being neither affine nor projective. Constructions and comparisons can therefore be organized by the chosen closure and complement rather than by an unrelated list of coordinate charts.

The compression does not supply a canonical embedding, preserve every equation of a chosen presentation, or make locally affine schemes automatically quasi-projective. Different immersions can expose different boundaries and line bundles. The property guarantees access to projective ambient methods while leaving the particular embedding, base field, and scheme-theoretic subtleties explicit.

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. Failure to produce the presentation is not by itself a proof of nonexistence, but merely showing that every point has an affine neighborhood does not establish the stronger embedding property.

The concept also organizes intervention and boundary reasoning. Passing from a projective variety to a Zariski-open subvariety preserves quasi-projectivity even though projectivity may be lost; embedding an affine variety in the standard affine open of projective space likewise proves quasi-projectivity without making the variety projective. Removing a point from projective space of dimension at least two predicts a representative intermediate case—quasi-projective but neither affine nor projective. These moves place affine and projective varieties inside the quasi-projective class while preventing the converse inference, and they identify the decisive boundary: the existence of a locally closed projective embedding, not resemblance to projective geometry or local affineness alone.

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. This vocabulary lets geometers move between affine, projective, and intermediate examples while keeping the chosen ambient space and removed boundary available for later constructions.

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. What transfers is locally closed embedding reasoning; what remains home-bound is the Zariski topology, algebraic variety or scheme, projective space, and the associated morphisms. Calling something “almost projective” or merely locally affine is only (A) analogy and supplies no quasi-projective certificate. The transfer stops where no locally closed projective immersion exists, and the property does not carry a canonical embedding or make different presentations preserve the same visible boundary.

Examples¶

Canonical¶

Let X = A¹ \ {0} over a field. In projective coordinates [X₀:X₁] on P¹, the affine line is the open chart X₀ ≠ 0, and removing its zero point [1:0] leaves the open set D₊(X₀X₁). Thus X is open—and hence locally closed—in the projective variety P¹, directly certifying quasi-projectivity.^[7] The same variety also has an affine presentation: x ↦ (x, x⁻¹) identifies it with the closed hyperbola xy − 1 = 0 in A².^[8]

Mapped back: X is candidate variety, P¹ is projective ambient space, and the inclusion of D₊(X₀X₁) supplies locally closed immersion. The set D₊(X₀X₁) is Zariski-open part, while all of P¹ can serve as Zariski-closed part; their intersection supplies intersection certificate. The projective line is projective closure, the hyperbola presentation realizes affine branch, and either presentation yields classification guarantee.

Applied / In Practice¶

A standard intermediate case is P² \ {p} for a single point p.^[9] It is quasi-projective because it is the Zariski-open complement of the closed point inside the projective plane. Yet this complement is neither affine nor projective: deleting the point destroys projectivity, while the resulting open variety is not affine in dimension at least two.^[10] The example therefore shows that quasi-projective is a genuine class between its two familiar subfamilies, not just a collective label for affine and projective varieties.

Mapped back: P² \ {p} is candidate variety and P² is projective ambient space as well as projective closure. The complement of p is Zariski-open part, all of P² supplies Zariski-closed part, and inclusion gives locally closed immersion with intersection certificate. Because the candidate lies in neither narrower subclass, it instantiates intermediate branch. The chosen deleted point also illustrates presentation limitation: the displayed complement belongs to this presentation rather than being canonical data of quasi-projectivity itself.

Structural Tensions¶

T1: Existence of an embedding versus noncanonical presentation. One locally closed immersion is enough to prove quasi-projectivity, but different projective embeddings can expose different equations, closures, and boundary complements. Diagnostic: Treat the existence of X = U ∩ Z as intrinsic to the classification while keeping the chosen U, Z, and ambient coordinates presentation-dependent.

T2: Affine-projective inclusion versus intermediate identity. The class gains usefulness by containing both affine and projective varieties, yet reducing it to their union erases cases that are neither. Diagnostic: Test the locally closed projective embedding directly, then separately ask whether the open or closed part places the variety in either narrower subclass.

T3: Local affineness versus global embeddability. Affine neighborhoods make calculation possible point by point, but they do not alone assemble into the required global projective immersion. Diagnostic: Require a global locally closed certificate rather than inferring quasi-projectivity from a cover by affine charts.

T4: Projective techniques versus boundary dependence. A projective closure lets an open variety inherit powerful ambient methods, while conclusions involving the removed locus may change with the compactification. Diagnostic: Separate statements invariant under the variety from statements that depend on the selected closure or boundary divisor.

T5: Quasi-projective-variety autonomy versus reduction to Pattern. Every qualifying quasi-projective variety is a strict algebraic-geometric specialization of the exact parent Prime Pattern (Pattern): across admissible presentations, the carrier repeatedly exhibits the invariant locally closed relation X = U ∩ Z or an equivalent open immersion into a projective closure. Reduction preserves that organized open–closed arrangement and recognition certificate, but loses the Zariski topology, scheme or variety carrier, projective ambient space, and local-affineness boundary. Treating quasi-projectivity as wholly autonomous hides its complete pattern structure.
Diagnostic: Is there merely a stable open–closed arrangement, or does a valid locally closed projective immersion establish the exact quasi-projective identity?

Structural–Framed Character¶

Quasi-Projective Variety is structural-leaning. Its locally closed embedding condition is exact and presentation-invariant at the level of existence, while its literal identity remains tied to varieties or schemes, the Zariski topology, and projective ambient space.

Its evaluative_weight is low: quasi-projectivity is a classification property rather than a claim of geometric quality. Its human_practice_bound is low because geometers choose a presentation and notation, but the existence of a valid immersion is not sustained by ongoing practice. Its institutional_origin is low; mathematical convention fixes terms and axioms without constituting the particular embedding relation. Its vocab_travels is low because “locally closed” and “projective” must retain their algebraic-geometric meanings for literal use. Its import_vs_recognize balance favors recognition: a chosen projective presentation can witness a pre-existing property, even though different witnesses expose different complements.

The smallest reviewed portable skeleton is Pattern (Pattern). Across admissible changes of presentation, the recurring open–closed organization and its locally closed certificate remain invariant, and the claim collapses when only local affineness or visual resemblance survives. That portable reach belongs to the Pattern Prime. Zariski openness and closedness, projective immersion, scheme structure, and affine/projective/intermediate branches remain the algebraic-geometric accent owned by Quasi-Projective Variety.

Its character: structural-leaning because an exact invariant embedding pattern governs membership while the carrier and topology sharply bound its literal recognition.

Structural Core vs. Domain Accent¶

Quasi-Projective Variety remains domain-specific rather than a Prime because its portable repeatable organization is constituted by an exact locally closed immersion in Zariski projective geometry.

What is skeletal (could lift toward a cross-domain prime). The complete thin skeleton is a typed carrier and scale, a repeatable organizing relation, admissible variations under which it persists, an invariant recognition criterion, an observation or proof map, and a collapse case that preserves surface resemblance while removing the relation. This candidate strictly instantiates Pattern: across equations, dimensions, and projective presentations, the locally closed organization X = U ∩ Z or equivalent open immersion recurs, while failure to exhibit that relation defeats quasi-projectivity. The Pattern is the organization itself, not the act of recognizing it.

What is domain-bound. The algebraic-geometric accent comprises a variety or scheme, projective ambient space, Zariski-open and Zariski-closed loci, locally closed immersion, projective closure, and the affine, projective, and intermediate branches. Isomorphism and change of presentation may preserve the classification, while the displayed complement and embedding need not be canonical; local affineness alone is insufficient.

Why this does not clear the prime bar. This complete variety–Zariski topology–projective ambient–locally closed immersion signature does not recur literally across three unrelated domains—a temporal signal motif, a grammatical construction, and a recurring software structure. Those domains can preserve carrier, repeated relation, admissible variation, invariant, and collapse test and thereby instantiate Pattern, but they do not thereby define a Quasi-Projective Variety; the portable reach belongs to Pattern. Remove the algebraic-geometric accent and the residue is a repeatable open–closed organization, not this candidate. Preserve the specialist nouns of variety, open, and projective but remove the locally closed immersion relation, and the residue is geometric vocabulary rather than a Quasi-Projective Variety.

This entry is a kind of Pattern.

Instantiates — Pattern (Pattern). The carriers are algebraic varieties or schemes considered at the granularity of an immersion into projective space. Across different equations, fields, embeddings, and dimensions, the same organizing relation recurs: Quasi-projective variety is a locally closed subobject, equivalently an intersection (U\cap Z) of a Zariski-open locus and a Zariski-closed locus, or an open subobject of a projective closure. Isomorphism and change of projective presentation are admissible variations; existence of that locally closed immersion is the invariant, and an exhibited immersion or closure is the proof-level observation map. Affine and projective varieties occupy limiting branches, while intermediate examples show that the relation is not accidental overlap between those classes. A positive test identifies carrier, scale, repeated open–closed organization, admissible presentation changes, certificate, and exact counterexample. A collapse test leaves local affineness without a global projective immersion, an arbitrary subset of projective space, or informal resemblance to projectivity. Replacing the algebraic-geometric terms by those typed roles preserves Pattern's complete signature, whereas deleting the locally closed relation destroys quasi-projectivity even if a projective ambient object remains. Quasi-projective variety is therefore a strict algebraic-geometric specialization of Pattern, with the Zariski topology and immersion structure as its domain accent.

Relationships to Other Abstractions¶

Current abstraction Quasi-projective variety Domain-specific

Parents (1) — more general patterns this builds on

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 (66^th 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

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Projective variety. A projective variety is closed in projective space, while a quasi-projective variety may be only locally closed or a proper open subset of its projective closure. Tell: check whether the chosen immersion has a nonempty removed boundary or is already projectively closed.
Affine variety. An affine variety is closed in affine space and forms a subtype of the quasi-projective class through the standard open embedding of affine space into projective space. Tell: establish an affine closed presentation rather than treating every locally closed projective presentation as affine.
Quasi-affine variety. A quasi-affine variety is an open subvariety of an affine variety and thereby a narrower subclass of quasi-projective varieties; not every quasi-projective variety admits the stronger open-in-affine presentation. Tell: prove the open-in-affine condition for quasi-affineness rather than stopping at the broader locally closed immersion into projective space that establishes quasi-projectivity.
Projective closure. The projective closure is the closed ambient variety obtained by adding a boundary to a chosen embedding; the original quasi-projective variety is its open part, not the closure itself. Tell: identify whether the boundary points are included in the object being classified.
Locally affine variety. Local affineness provides affine neighborhoods and is a broader local condition, whereas quasi-projectivity requires one global locally closed projective immersion. Tell: ask for the global embedding (U\cap Z), not merely an affine cover.

References¶

[1] The Stacks Project Authors, Quasi-projective Schemes, Tag 0B41 (accessed 2026-09-13). registry ↩ Show verification details

[2] Unverified encyclopedia synthesis; no authoritative source located for the claim as written. ↩

[3] Unverified encyclopedia synthesis; no authoritative source located for the claim as written. ↩

[4] Unverified encyclopedia synthesis; no authoritative source located for the claim as written. ↩

[5] Unverified encyclopedia synthesis; no authoritative source located for the claim as written. ↩

[6] Unverified encyclopedia synthesis; no authoritative source located for the claim as written. ↩

[7] Unverified encyclopedia synthesis; no authoritative source located for the claim as written. ↩

[8] Unverified encyclopedia synthesis; no authoritative source located for the claim as written. ↩

[9] Unverified encyclopedia synthesis; no authoritative source located for the claim as written. ↩

[10] Unverified encyclopedia synthesis; no authoritative source located for the claim as written. ↩