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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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283:: 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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164:; similarly for projective varieties. For example, the complement of a point in the affine line, i.e.,
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160:. 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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204:{\displaystyle X=\mathbb {A} ^{1}\setminus \{0\}}
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107:can be expressed as an intersection of the
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55:subset. A similar definition is used in
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240:in the affine plane. As an affine set
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75:Relationship to affine varieties
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143:is quasiprojective. There are
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83:is a Zariski-open subset of a
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329:Encyclopedia of Mathematics
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358:Basic Algebraic Geometry 1
297:Abstract algebraic variety
132:{\displaystyle {\bar {U}}}
371:10.1007/978-3-642-37956-7
324:"Quasi-projective scheme"
275:in the same sense that a
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109:projective completion
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233:{\displaystyle xy-1}
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63:is a locally closed
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303:divisorial scheme
253:{\displaystyle X}
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279:is locally
29:mathematics
345:References
59:, where a
334:EMS Press
310:Citations
281:Euclidean
225:−
190:∖
158:varieties
124:¯
65:subscheme
393:Category
355:(2013).
291:See also
277:manifold
152:Examples
67:of some
336:, 2001
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262:conic
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