Difference between revisions of "Quasi-projective scheme"
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| − | A locally closed subscheme of a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076660/q0766601.png" />. In other words, a quasi-projective scheme is an open subscheme of a [[Projective scheme|projective scheme]]. A [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076660/q0766602.png" /> over a field is quasi-projective if and only if there exists on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076660/q0766603.png" /> an invertible [[Ample sheaf|ample sheaf]]. A generalization of the notion of a | + | A locally closed subscheme of a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076660/q0766601.png" />. In other words, a quasi-projective scheme is an open subscheme of a [[Projective scheme|projective scheme]]. A [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076660/q0766602.png" /> over a field is quasi-projective if and only if there exists on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076660/q0766603.png" /> 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 composite of an open imbedding and a projective morphism. A scheme that is both quasi-projective and complete is projective. |
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Revision as of 21:55, 30 March 2012
A locally closed subscheme of a projective space 
. In other words, a quasi-projective scheme is an open subscheme of a projective scheme. A scheme 
over a field is quasi-projective if and only if there exists on 
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 composite of an open imbedding and a projective morphism. A scheme that is both quasi-projective and complete is projective.
Comments
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=18339
Quasi-projective scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-projective_scheme&oldid=18339
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article