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Difference between revisions of "Intersection index (in algebraic geometry)"

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The number of points in the intersection of  $  n $
 
The number of points in the intersection of  $  n $
divisors (cf. [[Divisor|Divisor]]) in an  $  n $-
+
divisors (cf. [[Divisor|Divisor]]) in an  $  n $-dimensional [[Algebraic variety|algebraic variety]] with allowance for the multiplicities of these points. More precisely, let  $  X $
dimensional [[Algebraic variety|algebraic variety]] with allowance for the multiplicities of these points. More precisely, let  $  X $
+
be an  $  n $-dimensional non-singular algebraic variety over a field  $  k $,  
be an  $  n $-
 
dimensional non-singular algebraic variety over a field  $  k $,  
 
 
and let  $  D _ {1} \dots D _ {n} $
 
and let  $  D _ {1} \dots D _ {n} $
 
be effective divisors in  $  X $
 
be effective divisors in  $  X $
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$$  
 
$$  
( D _ {1} \dots D _ {n} ) _ {x}  =  \mathop{\rm dim} _ {k}  A / ( u _ {1} \dots
+
( D _ {1} \dots D _ {n} ) _ {x}  =  \mathop{{\rm dim}_{k}}  A / ( u _ {1} \dots u _ {n} ) ,
u _ {n} ) ,
 
 
$$
 
$$
  
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is the local equation for the divisor  $  D _ {i} $
 
is the local equation for the divisor  $  D _ {i} $
 
in the local ring  $  A = {\mathcal O} _ {X,x} $.  
 
in the local ring  $  A = {\mathcal O} _ {X,x} $.  
In the complex case, the local index coincides with the residue of the form  $  ( du _ {1} /u _ {1} ) \wedge \dots \wedge ( du _ {n} /u _ {n} ) $,  
+
In the complex case, the local index coincides with the residue of the form  $  ( du _ {1} /u _ {1} ) \wedge \dots \wedge ( du _ {n} /u _ {n} ) $, and also with the degree of the germ of the mapping (cf. [[Degree of a mapping|Degree of a mapping]])
and also with the degree of the germ of the mapping (cf. [[Degree of a mapping|Degree of a mapping]])
 
  
 
$$  
 
$$  

Latest revision as of 06:42, 29 December 2021


The number of points in the intersection of $ n $ divisors (cf. Divisor) in an $ n $-dimensional algebraic variety with allowance for the multiplicities of these points. More precisely, let $ X $ be an $ n $-dimensional non-singular algebraic variety over a field $ k $, and let $ D _ {1} \dots D _ {n} $ be effective divisors in $ X $ that intersect in a finite number of points. The local index (or multiplicity) of intersection of these divisors at a point $ x \in X $ is the integer

$$ ( D _ {1} \dots D _ {n} ) _ {x} = \mathop{{\rm dim}_{k}} A / ( u _ {1} \dots u _ {n} ) , $$

where $ u _ {i} $ is the local equation for the divisor $ D _ {i} $ in the local ring $ A = {\mathcal O} _ {X,x} $. In the complex case, the local index coincides with the residue of the form $ ( du _ {1} /u _ {1} ) \wedge \dots \wedge ( du _ {n} /u _ {n} ) $, and also with the degree of the germ of the mapping (cf. Degree of a mapping)

$$ ( u _ {1} \dots u _ {n} ): ( X, x) \rightarrow ( \mathbf C ^ {n} , 0). $$

The global intersection index $ ( D _ {1} \dots D _ {n} ) $ is the sum of the local indices over all points of the intersection $ D _ {1} \cap {} \dots \cap D _ {n} $. If this intersection is not empty, then $ ( D _ {1} \dots D _ {n} ) > 0 $.

See also Intersection theory.

Comments

References

[a1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1974) pp. Chapt. IV (Translated from Russian) MR0366917 Zbl 0284.14001
How to Cite This Entry:
Intersection index (in algebraic geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intersection_index_(in_algebraic_geometry)&oldid=51798
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article