Let $X$ be a scheme. An ideal in ${\mathcal{O}}_{X}$ is a quasi-coherent ideal sheaf in ${\mathcal{O}}_{X}$. A divisor on $X$ is always a Cartier divisor on $X$. Let $k$ be a field. A variety $X$ over $k$ is an integral $k$-scheme $X$ which is separated and of finite type. A curve (resp. surface) is a variety of dimension one (resp. two).

Throughout this paper $(K,{|\phantom{a}|})$ denotes a complete non-archimedean valued field with valuation ring $K^{\circ}$ and residue field $\tilde{K}$. Starting in Section 7 we will assume furthermore that the valuation is discrete and that $K$ has positive characteristic $p>0$. In this case there exists an isomorphism $K^{\circ}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\tilde{K}[[T]]$ [Mat89, Thm. 29.7]. Let $X$ be a $K$-variety. We denote the analytification of $X$ in the sense of Berkovich [Ber90, Thm. 3.4.1] by $X^{\mathrm{an}}$.