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2 and higher. While any complete nonsingular surface is projective, there exist nonsingular complete varieties in dimension 3 and higher which are not projective. The first examples of non-projective complete varieties were given by
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158:. An intuitive justification of "complete", in the sense of "no missing points", can be given on the basis of the
443:
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251:
Zariski, Oscar (1958). "Introduction to the
Problem of Minimal Models in the Theory of Algebraic Surfaces".
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is compact if and only if the above projection map is closed with respect to topological products.
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The image of a complete variety is closed and is a complete variety. A closed
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406:, Lecture Notes in Mathematics, vol. 1358 (Second, expanded ed.),
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on it will have more closed sets (except in very simple cases). See also
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291:"Existence theorems for nonprojective complete algebraic varieties"
459:. Graduate Texts in Mathematics. Vol. 52. Berlin, New York:
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A complex variety is complete if and only if it is compact as a
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The morphism taking a complete variety to a point is a
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onto closed sets). This can be seen as an analogue of
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122:The most common example of a complete variety is a
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453:(1977). "Appendix B. Example 3.4.1. (Fig.24)".
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147:of positive dimension is not complete.
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325:On the theory of birational blowing-up
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404:The red book of varieties and schemes
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112:of a complete variety is complete.
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160:valuative criterion of properness
128:complete non-projective varieties
253:American Journal of Mathematics
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76:{\displaystyle X\times Y\to Y}
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1:
356:Graduate Texts in Mathematics
328:(thesis). Harvard University.
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434:"Complete algebraic variety"
38:, such that for any variety
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439:Encyclopedia of Mathematics
169:
10:
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322:Hironaka, Heisuke (1960).
289:Nagata, Masayoshi (1958).
29:complete algebraic variety
98:in algebraic geometry: a
16:Type of algebraic variety
381:Milne, James S. (2009),
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117:complex-analytic variety
432:Danilov, V.I. (2001) ,
308:10.1215/ijm/1255454111
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162:, which goes back to
126:, but there do exist
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505:Algebraic varieties
218:does not carry the
181:Theorem of the cube
23:, in particular in
456:Algebraic Geometry
384:Algebraic geometry
351:Algebraic Geometry
222:, in general; the
154:, in the sense of
124:projective variety
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25:algebraic geometry
470:978-0-387-90244-9
451:Hartshorne, Robin
425:978-3-540-63293-1
365:978-0-387-90244-9
346:Hartshorne, Robin
100:topological space
33:algebraic variety
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343:Section II.4 of
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295:Illinois J. Math
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224:Zariski topology
220:product topology
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164:Claude Chevalley
141:Heisuke Hironaka
137:Masayoshi Nagata
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397:Section I.9 of
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379:Chapter 7 of
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186:Fano variety
176:Chow's lemma
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145:affine space
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301:: 490–498.
259:: 146–184.
96:compactness
92:closed sets
90:(i.e. maps
21:mathematics
487:0367.14001
391:2010-08-04
238:References
132:dimensions
110:subvariety
88:closed map
44:projection
444:EMS Press
387:, v. 5.20
204:Here the
68:→
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499:Category
402:(1999),
348:(1977),
208:variety
170:See also
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479:0463157
374:0463157
337:Sources
273:2372827
206:product
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31:is an
269:JSTOR
192:Notes
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