Difference between revisions of "Quasi-projective scheme"
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| − | A locally closed | + | A locally closed sub-scheme of a projective space $ \mathbf{P}^{n} $. In other words, a ''quasi-projective scheme'' is an open sub-scheme of a [[Projective scheme|projective scheme]]. A [[Scheme|scheme]] $ X $ over a field is quasi-projective if and only if there exists on $ X $ an invertible [[Ample sheaf|ample sheaf]]. A generalization of the notion of a quasi-projective scheme is that of a ''quasi-projective morphism'', that is, a [[Morphism|morphism]] of schemes that is the composition of an open imbedding and a projective morphism. A scheme that is both quasi-projective and complete is projective. |
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, | + | <table> |
| + | <TR><TD valign="top">[a1]</TD><TD valign="top"> R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 10 & 103. {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR> | ||
| + | </table> | ||
Latest revision as of 06:48, 10 May 2016
A locally closed sub-scheme of a projective space $ \mathbf{P}^{n} $. In other words, a quasi-projective scheme is an open sub-scheme of a projective scheme. A scheme $ X $ over a field is quasi-projective if and only if there exists on $ X $ an invertible ample sheaf. A generalization of the notion of a quasi-projective scheme is that of a quasi-projective morphism, that is, a morphism of schemes that is the composition of an open imbedding and a projective morphism. A scheme that is both quasi-projective and complete is projective.
References
| [a1] | R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 10 & 103. MR0463157 Zbl 0367.14001 |
How to Cite This Entry:
Quasi-projective scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-projective_scheme&oldid=23946
Quasi-projective scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-projective_scheme&oldid=23946
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article