Let $h\in\PA(\Delta_{\mathcal{X}})_{\mathbf{Z}}$. For each $J\subset I$ such that $E_{J}\neq\emptyset$ we set $\mathcal{X}_{J}:=\mathcal{X}\setminus\bigcup_{i\in I\setminus J}E_{i}$ and define a vertical fractional ideal sheaf $\mathfrak{a}_{h}$ on $\mathcal{X}$ by letting for each $J$