View source for Quasi-projective variety

In [[mathematics]], a '''quasi-projective variety''' in [[algebraic geometry]] is a [[locally closed subset]] of a [[projective variety]], i.e., the intersection inside some [[projective space]] of a Zariski-open and a [[Zariski topology|Zariski-closed]] subset. A similar definition is used in [[Scheme (mathematics)|scheme theory]], where a ''quasi-projective scheme'' is a locally closed [[subscheme]] of some [[Algebraic geometry of projective spaces|projective space]].<ref>{{SpringerEOM|title=Quasi-projective scheme}}</ref>

==Relationship to affine varieties==

An [[affine space]] is a Zariski-open subset of a [[projective space]], and since any closed affine subset <math>U</math> can be expressed as an intersection of the [[Homogeneous polynomial#Homogenization|projective completion]] <math>\bar{U}</math> and the affine space embedded in the projective space, this implies that any [[affine variety]] is quasiprojective. There are [[locally closed]] subsets of projective space that are not affine, so that quasi-projective is more general than affine. Taking the complement of a single point in projective space of dimension at least 2 gives a non-affine quasi-projective variety. This is also an example of a quasi-projective variety that is neither affine nor projective.

==Examples==

Since quasi-projective varieties generalize both affine and projective varieties, they are sometimes referred to simply as ''varieties''. Varieties isomorphic to affine algebraic varieties as quasi-projective varieties are called [[affine variety|affine varieties]]; similarly for projective varieties. For example, the complement of a point in the affine line, i.e., <math>X=\mathbb{A}^1 \setminus \{0\}</math>, is isomorphic to the zero set of the polynomial <math>xy-1</math> in the affine plane. As an affine set <math>X</math> is not closed since any polynomial zero on the complement must be zero on the affine line. For another example, the complement of any [[Conic section|conic]] in projective space of dimension 2 is affine. Varieties isomorphic to open subsets of affine varieties are called '''quasi-affine'''.

Quasi-projective varieties are ''locally affine'' in the same sense that a [[manifold]] is locally [[Euclidean space|Euclidean]]: every point of a quasi-projective variety has a neighborhood which is an affine variety. This yields a basis of affine sets for the [[Zariski topology]] on a quasi-projective variety.

== See also ==
*[[Abstract algebraic variety]], sometimes synonymous with "quasi-projective variety"
*[[divisorial scheme]], a generalization of a quasi-projective variety

== Citations ==
{{Reflist}}

== References ==
{{refbegin}}
* {{cite book | last=Shafarevich | first=Igor R. | author-link=Igor Shafarevich| title = Basic Algebraic Geometry 1 | year=2013| publisher=[[Springer Science+Business Media|Springer Science]] | doi=10.1007/978-3-642-37956-7 | url=https://link.springer.com/book/10.1007/978-3-642-37956-7 | isbn=978-0-387-97716-4}}
{{refend}}

[[Category:Algebraic varieties]]