|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| 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>
| | Nice to satisfy you, my name is Ling and I completely dig that name. Her family life in Idaho. His day job is a cashier and his salary has been truly fulfilling. One of his preferred hobbies is playing crochet but he hasn't produced a dime with it.<br><br>Here is my site - car warranty - [http://trochoi.vn/profile/312428/ChMullagh.html mouse click the next webpage] - |
| | |
| ==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}^1-0</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 — 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]]
| |
Latest revision as of 11:43, 14 December 2014
Nice to satisfy you, my name is Ling and I completely dig that name. Her family life in Idaho. His day job is a cashier and his salary has been truly fulfilling. One of his preferred hobbies is playing crochet but he hasn't produced a dime with it.
Here is my site - car warranty - mouse click the next webpage -