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

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

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

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

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