Difference between revisions of "Intersection index (in algebraic geometry)"
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Revision as of 21:53, 30 March 2012
The number of points in the intersection of 
divisors (cf. Divisor) in an 
-dimensional algebraic variety with allowance for the multiplicities of these points. More precisely, let 
be an 
-dimensional non-singular algebraic variety over a field 
, and let 
be effective divisors in 
that intersect in a finite number of points. The local index (or multiplicity) of intersection of these divisors at a point 
is the integer
 |
where 
is the local equation for the divisor 
in the local ring 
. In the complex case, the local index coincides with the residue of the form 
, and also with the degree of the germ of the mapping (cf. Degree of a mapping)
 |
The global intersection index 
is the sum of the local indices over all points of the intersection 
. If this intersection is not empty, then 
.
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 |
Intersection index (in algebraic geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intersection_index_(in_algebraic_geometry)&oldid=18849