# Difference between revisions of "Invertible sheaf"

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An invertible sheaf on a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250035.png" /> is related to rational mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250036.png" /> into projective spaces. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250037.png" /> be an invertible sheaf on a scheme and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250038.png" /> be sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250039.png" /> the values of which at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250040.png" /> generate the stalk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250041.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250042.png" />. Then there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250046.png" /> are homogeneous coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250047.png" />. An invertible sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250048.png" /> is called very ample if there exists an imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250050.png" />. An invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250051.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250052.png" /> is called ample if there exists a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250053.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250054.png" /> is very ample. On a [[Noetherian scheme|Noetherian scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250055.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250056.png" /> an invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250057.png" /> is ample if and only if for each [[Coherent sheaf|coherent sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250059.png" /> there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250060.png" /> such that the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250061.png" /> is generated by its global sections for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250062.png" />. | An invertible sheaf on a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250035.png" /> is related to rational mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250036.png" /> into projective spaces. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250037.png" /> be an invertible sheaf on a scheme and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250038.png" /> be sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250039.png" /> the values of which at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250040.png" /> generate the stalk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250041.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250042.png" />. Then there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250046.png" /> are homogeneous coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250047.png" />. An invertible sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250048.png" /> is called very ample if there exists an imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250050.png" />. An invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250051.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250052.png" /> is called ample if there exists a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250053.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250054.png" /> is very ample. On a [[Noetherian scheme|Noetherian scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250055.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250056.png" /> an invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250057.png" /> is ample if and only if for each [[Coherent sheaf|coherent sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250059.png" /> there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250060.png" /> such that the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250061.png" /> is generated by its global sections for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250062.png" />. | ||

− | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250063.png" /> is an ample invertible sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250064.png" /> corresponding to a divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250066.png" /> is called an ample divisor. A Cartier divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250067.png" /> on a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250068.png" /> that is proper and smooth over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250069.png" /> is ample if and only if for each closed integral subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250070.png" /> the intersection index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250071.png" /> is positive, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250072.png" /> (cf. [[Intersection index (in algebraic geometry)|Intersection index (in algebraic geometry)]]). For other criteria of ampleness see [[#References|[5]]]. There is also a generalization of the concept of an ample divisor on subvarieties of large codimension [[#References|[2]]]. | + | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250063.png" /> is an ample invertible sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250064.png" /> corresponding to a divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250066.png" /> is called an [[ample divisor]]. A Cartier divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250067.png" /> on a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250068.png" /> that is proper and smooth over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250069.png" /> is ample if and only if for each closed integral subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250070.png" /> the intersection index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250071.png" /> is positive, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250072.png" /> (cf. [[Intersection index (in algebraic geometry)|Intersection index (in algebraic geometry)]]). For other criteria of ampleness see [[#References|[5]]]. There is also a generalization of the concept of an ample divisor on subvarieties of large codimension [[#References|[2]]]. |

The concepts of very ample and ample invertible sheaves can be carried over to the case of analytic spaces (for criteria for ampleness in this situation see [[Positive vector bundle|Positive vector bundle]]). | The concepts of very ample and ample invertible sheaves can be carried over to the case of analytic spaces (for criteria for ampleness in this situation see [[Positive vector bundle|Positive vector bundle]]). |

## Revision as of 21:19, 20 December 2014

A locally free sheaf of -modules of rank 1 on a ringed space . An equivalent definition is: A sheaf of -modules that is locally isomorphic to the sheaf . The invertible sheaves on , considered up to isomorphism, form an Abelian group with respect to the operation of tensor multiplication over . This group is called the Picard group of the space , and is denoted by . The inverse of a sheaf in this group is the sheaf dual to . In the case when is a scheme (in particular, an algebraic variety) or an analytic space, a sheaf of -modules is invertible if and only if it is isomorphic to the sheaf of regular (respectively, analytic) sections of some algebraic (respectively, analytic) line bundle over .

Invertible sheaves on schemes are closely connected with divisors (cf. Divisor). With each Cartier divisor on is associated an invertible sheaf , thus defining an injective homomorphism , where is the group of classes of Cartier divisors on . For integral schemes this homomorphism is an isomorphism.

On a projective scheme Serre's twisted invertible sheaf can be defined. In fact, if an imbedding of the scheme in a projective space is given, then corresponds to the class of a hyperplane section. In particular, if is a projective space over a field , then the sheaf is the direct image of the sheaf of linear functions on under the natural mapping . The system of homogeneous coordinates in can be identified with a basis for the space of sections .

An invertible sheaf on a scheme is related to rational mappings of into projective spaces. Let be an invertible sheaf on a scheme and let be sections of the values of which at any point generate the stalk over . Then there exists a unique morphism such that and , where are homogeneous coordinates in . An invertible sheaf on is called very ample if there exists an imbedding such that . An invertible sheaf on is called ample if there exists a positive integer for which is very ample. On a Noetherian scheme over an invertible sheaf is ample if and only if for each coherent sheaf on there exists an integer such that the sheaf is generated by its global sections for .

If is an ample invertible sheaf on corresponding to a divisor , then is called an ample divisor. A Cartier divisor on a scheme that is proper and smooth over an algebraically closed field is ample if and only if for each closed integral subscheme the intersection index is positive, where (cf. Intersection index (in algebraic geometry)). For other criteria of ampleness see [5]. There is also a generalization of the concept of an ample divisor on subvarieties of large codimension [2].

The concepts of very ample and ample invertible sheaves can be carried over to the case of analytic spaces (for criteria for ampleness in this situation see Positive vector bundle).

#### References

[1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |

[2] | R. Hartshorne, "Ample subvarieties of algebraic varieties" , Springer (1970) MR0282977 Zbl 0208.48901 |

[3] | D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701 |

[4] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |

[5] | I.V. Dolgachev, "Abstract algebraic geometry" J. Soviet Math. , 2 : 3 (1974) pp. 264–303 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 10 (1972) pp. 47–112 Zbl 1068.14059 |

#### Comments

The definition of Serre's twisted invertible sheaf is not precise enough. There is an action of the multiplicative group on which has as its quotient. The direct image of the structure sheaf under the mapping splits into a direct sum of invertible sheaves , , such that acts on via the character .

**How to Cite This Entry:**

Invertible sheaf.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Invertible_sheaf&oldid=23868