Intersection product of Cartier divisors ($D_1\cdots D_n\cdot [X]$)

Successive intersection with Cartier divisors gives a cycle class. On an $n$-dimensional complete scheme, its degree is multilinear and depends only on numerical divisor classes; the top product with one divisor is its self-intersection number.

For a proper morphism of varieties $f:X\to Y$, a Cartier divisor $D$ on $Y$, and an algebraic cycle $\alpha$ on $X$, pullback of $D$, intersection with $\alpha$, and proper pushforward satisfy the displayed identity in the Chow group. In particular, for a generically finite morphism of surfaces of degree $r$, $(f^{\ast }D{)}^2=r{D}^2$. For a curve $C$ contracted to a point, $f_{\ast }C=0$, so $f^{\ast }D\cdot C=0$.