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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
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272:: every point of a quasi-projective variety has a neighborhood which is an affine variety. This yields a basis of affine sets for the
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is not closed since any polynomial zero on the complement must be zero on the affine line. For another example, the complement of any
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Since quasi-projective varieties generalize both affine and projective varieties, they are sometimes referred to simply as
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in projective space of dimension 2 is affine. Varieties isomorphic to open subsets of affine varieties are called
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153:; similarly for projective varieties. For example, the complement of a point in the affine line, i.e.,
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149:. Varieties isomorphic to affine algebraic varieties as quasi-projective varieties are called
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and the affine space embedded in the projective space, this implies that any
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193:{\displaystyle X=\mathbb {A} ^{1}\setminus \{0\}}
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96:can be expressed as an intersection of the
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44:subset. A similar definition is used in
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229:in the affine plane. As an affine set
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64:Relationship to affine varieties
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260:Quasi-projective varieties are
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132:is quasiprojective. There are
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72:is a Zariski-open subset of a
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318:Encyclopedia of Mathematics
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347:Basic Algebraic Geometry 1
286:Abstract algebraic variety
121:{\displaystyle {\bar {U}}}
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313:"Quasi-projective scheme"
264:in the same sense that a
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222:{\displaystyle xy-1}
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292:divisorial scheme
242:{\displaystyle X}
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268:is locally
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334:References
48:, where a
323:EMS Press
299:Citations
270:Euclidean
214:−
179:∖
147:varieties
113:¯
54:subscheme
382:Category
344:(2013).
280:See also
266:manifold
141:Examples
56:of some
325:, 2001
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251:conic
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110:U
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