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The number of points in the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i0520101.png" /> divisors (cf. [[Divisor|Divisor]]) in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i0520102.png" />-dimensional [[Algebraic variety|algebraic variety]] with allowance for the multiplicities of these points. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i0520103.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i0520104.png" />-dimensional non-singular algebraic variety over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i0520105.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i0520106.png" /> be effective divisors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i0520107.png" /> that intersect in a finite number of points. The local index (or multiplicity) of intersection of these divisors at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i0520108.png" /> is the integer
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i0520109.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i05201010.png" /> is the local equation for the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i05201011.png" /> in the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i05201012.png" />. In the complex case, the local index coincides with the residue of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i05201013.png" />, and also with the degree of the germ of the mapping (cf. [[Degree of a mapping|Degree of a mapping]])
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The number of points in the intersection of  $  n $
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divisors (cf. [[Divisor|Divisor]]) in an  $  n $-dimensional [[Algebraic variety|algebraic variety]] with allowance for the multiplicities of these points. More precisely, let  $  X $
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be an  $  n $-dimensional non-singular algebraic variety over a field  $  k $,  
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and let  $  D _ {1} \dots D _ {n} $
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be effective divisors in  $  X $
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that intersect in a finite number of points. The local index (or multiplicity) of intersection of these divisors at a point  $  x \in X $
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is the integer
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i05201014.png" /></td> </tr></table>
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$$
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( D _ {1} \dots D _ {n} ) _ {x}  =   \mathop{{\rm dim}_{k}}  A / ( u _ {1} \dots u _ {n} ) ,
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$$
  
The global intersection index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i05201015.png" /> is the sum of the local indices over all points of the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i05201016.png" />. If this intersection is not empty, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052010/i05201017.png" />.
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where  $  u _ {i} $
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is the local equation for the divisor  $  D _ {i} $
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in the local ring  $  A = {\mathcal O} _ {X,x} $.  
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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]])
  
See also [[Intersection theory|Intersection theory]].
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$$
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( u _ {1} \dots u _ {n} ):  ( X, x)  \rightarrow  ( \mathbf C  ^ {n} , 0).
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$$
  
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The global intersection index  $  ( D _ {1} \dots D _ {n} ) $
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is the sum of the local indices over all points of the intersection  $  D _ {1} \cap {} \dots \cap D _ {n} $.
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If this intersection is not empty, then  $  ( D _ {1} \dots D _ {n} ) > 0 $.
  
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See also [[Intersection theory|Intersection theory]].
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.R. Shafarevich,   "Basic algebraic geometry" , Springer (1974) pp. Chapt. IV (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1974) pp. Chapt. IV (Translated from Russian) {{MR|0366917}} {{ZBL|0284.14001}} </TD></TR></table>

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=18849
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article