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Algebraic Geometry Intersection Number

Let Y and Z be subvarieties of complementary dimensions in a projective space P^n, and suppose that their intersection consists of finitely many points. Their algebraic geometry intersection number is

I(Y,Z)=sum_(p in Y intersection Z)i_p(Y,Z),

where i_p(Y,Z) is the local intersection multiplicity at p. A transversal intersection point has multiplicity 1, while a tangency can have multiplicity greater than 1. In the Chow ring, the number is the degree of the intersection product of the two cycle classes (Fulton 1998, Shafarevich 2013).

See also

Algebraic Geometry, Algebraic Variety, Chow Ring, Intersection Number, Multiplicity, Projective Space, Transversal Intersection

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References

Fulton, W. Intersection Theory, 2nd ed. New York: Springer-Verlag, 1998. https://doi.org/10.1007/978-1-4612-1700-8.Shafarevich, I. R. Basic Algebraic Geometry 1: Varieties in Projective Space, 3rd ed. Berlin, Germany: Springer-Verlag, 2013. https://doi.org/10.1007/978-3-642-37956-7.

Cite this as:

Weisstein, Eric W. "Algebraic Geometry Intersection Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlgebraicGeometryIntersectionNumber.html

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