Namespaces
Variants
Actions

Difference between revisions of "Quasi-projective scheme"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (TeX-ed article.)
 
Line 1: Line 1:
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.
+
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.
 
 
 
 
 
 
====Comments====
 
 
 
  
 
====References====
 
====References====
<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>
+
<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=38802
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article