Definition 9.10.2F30317C89D08

Definition 9.10.2.F30317C89D08 A prime divisor over an irreducible complex space $Z$ is a connected smooth hypersurface $E\subseteq X'$, where $X'\rightarrow Z$ is a proper bimeromorphic morphism with $X'$ smooth. Such a morphism $X'\rightarrow Z$ is also called a resolution of $Z$. The center of the prime divisor is defined as the image of $E$ in $Z$.

Two prime divisors $E_1\subseteq X'_1$ and $E_2\subseteq X'_2$ over $Z$ are equivalent if there is a common resolution $X''\rightarrow Z$ dominating both $X_1'$ and $X_2'$ such that the strict transforms of $E_1$ and $E_2$ coincide.

The set $Z^{\operatorname{div}}$ is the set of pairs $(c,E)$, where $c\in \mathbb {Q}_{>0}$ and $E$ is an equivalence class of a prime divisor over $Z$. For simplicity, we will denote the pair $(c,E)$ by $c\operatorname{ord}_ E$, although one should not really think of this object as a valuation unless $Z$ is projective and irreducible.