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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-closed]] subset. A similar definition is used in [[scheme theory]], where a ''quasi-projective scheme'' is a locally closed [[subscheme]] of some projective space.<ref>[http://eom.springer.de/q/q076660.htm Quasi-projective scheme - Encyclopedia of Mathematics]</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.
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==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}^1-0</math>, is isomorphic to the zero set of the polynomial <math>xy-1</math> in the affine plane. As an affine set X 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 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 sense that a [[manifold]] is locally Euclidean &mdash; every point of a quasiprojective variety has a neighborhood given by an affine variety. This yields a basis of affine sets for the Zariski topology on a quasi-projective variety.
 
== See also ==
*[[Abstract algebraic variety]], often synonymous with "quasi-projective variety"
 
== References ==
* Igor R. Shafarevich, ''Basic Algebraic Geometry 1'', Springer-Verlag 1999: Chapter 1 Section 4.
 
==Notes==
{{Reflist}}
 
[[Category:Algebraic varieties]]

Revision as of 03:12, 26 February 2014


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