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Bezout's Theorem

Definition. Suppose that $ X$ and $ Y\subset \mathbb{P}^n$ are two irreducible varieties and that their intersection has irreducible components $ Z_i$. We say that $ X$ and $ Y$ intersect generically transversally if, for each $ i$, $ X$ and $ Y$ intersect transversally at a general point $ p_i\in Z_i$, i.e., are smooth at $ p_i$ with tangent spaces spanning $ \mathbb{T}_{p_i}({\mathbb{P}}^n)$ (the tangent space to $ \mathbb{P}^n$ at $ p_i$.)

Theorem 5 (Bezout)   Let $ X$ and $ Y\subset \mathbb{P}^n$ be subvarieties of pure dimensions $ k$ and $ l$ with $ k+l\geq n$, and suppose they intersect generically transversely. Then

deg$\displaystyle \,(X\cap Y)=$deg$\displaystyle (X)\cdot$   deg$\displaystyle (Y).$

In particular, if $ k+l=n$, this says that $ X\cap Y$ will consist of deg$ (X)\cdot$   deg$ (Y)$ points.

A pair of pure-dimensional varieties $ X$ and $ Y\subset \mathbb{P}^n$ intersect properly if their intersection has the expected dimension, i.e.,

dim$\displaystyle (X\cap Y)=$dim$\displaystyle (X)+$dim$\displaystyle (Y)-n.$

Theorem 6   If $ X$ and $ Y$ intersect properly,

deg$\displaystyle (X)\cdot$   deg$\displaystyle (Y)=\sum m_Z(X,Y)\cdot$   deg$\displaystyle (Z)$

where the sum is over all irreducible subvarieties $ Z$ of the appropriate dimension (in effect, over all irreducible components $ Z$ of $ X\cap Y$). Here $ m_Z(X,Y)$ is the intersection multiplicity of $ X$ and $ Y$ along $ Z$:
1.
$ m_Z(X,Y)\geq 1$ for all $ Z\subset X\cap Y$ ( $ m_Z(X,Y)=0$ otherwise.)

2.
$ m_Z(X,Y)=1$ if $ X$ and $ Y$ intersect transversely at a general point of $ Z$.

3.
$ m_Z(X,Y)$ is additive, i.e. $ m_Z(X\cup X^{\prime},Y)=
m_Z(X,Y)+m_Z(X^{\prime},Y)$ for any $ X$ and $ X^{\prime}$ as long as all three numbers are defined and $ X$ and $ X^{\prime}$ have no common components.


In particular, for any subvarieties $ X$ and $ Y$ of pure dimension in $ \mathbb{P}^n$ intersecting properly:

deg$\displaystyle (X\cap Y)\leq$   deg$\displaystyle (X)\cdot$   deg$\displaystyle (Y).$


next up previous
Next: About this document ... Up: Algebraic Geometry Previous: Zariski Topology
Zvezdelina Stankova-Frenkel 2001-01-24