Bézout’s theorem (Algebraic Geometry)

The classic version of Bézout’s theorem states that two complex projective curves of degrees $m$ and $n$ which share no common component intersect in exactly $mn$ points if the points are counted with multiplicity.

The generalized version of Bézout’s theorem states that if $A$ and $B$ are algebraic varieties in $k$-dimensional projective space over an algebraically complete field and $A\cap B$ is a variety of dimension $\dim (A)+\dim (B)-k$, then the degree of $A\cap B$ is the product of the degrees of $A$ and $B$.