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Bezout's Theorem

Definition. Suppose that

and $Y\subset \mathbb{P}^n$ are two irreducible varieties and that their intersection has irreducible components

. We say that

and

intersect generically transversally if, for each

,

and

intersect transversally at a general point $p_i\in Z_i$ , i.e., are smooth at

with tangent spaces spanning $\mathbb{T}_{p_i}({\mathbb{P}}^n)$ (the tangent space to $\mathbb{P}^n$ at

.)

Theorem 5 (Bezout) Let

and $Y\subset \mathbb{P}^n$ be subvarieties of pure dimensions

and

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

, this says that $X\cap Y$ will consist of deg $(X)\cdot$ deg

points.

A pair of pure-dimensional varieties 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

and

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

of the appropriate dimension (in effect, over all irreducible components

of $X\cap Y$ ). Here

is the intersection multiplicity of

and

along

:

1.

$m_Z(X,Y)\geq 1$ for all $Z\subset X\cap Y$ (

otherwise.)

2.

if

and

intersect transversely at a general point of

.

3.

is additive, i.e. $m_Z(X\cup X^{\prime},Y)= m_Z(X,Y)+m_Z(X^{\prime},Y)$ for any

and $X^{\prime}$ as long as all three numbers are defined and

and $X^{\prime}$ have no common components.

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

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

Next: About this document ... Up: Algebraic Geometry Previous: Zariski Topology

Zvezdelina Stankova-Frenkel 2001-01-24