See Fulton’s Intersection Theory, Section 2.2.
Definition
A pseudo-divisor \(D\) on a scheme \(X\) is a triple \((L, Z, s)\).
- \(L\) is a line bundle on \(X\)
- \(Z\) is a closed subset of \(X\)
- \(s\) is a nowhere vanishing section of \(L\) on \(X-Z\); equivalently, it is a trivialization of \(L|_{X-Z}\).
We often denote this data as \((\mathcal O_X(D), |D|, s_D)\).