This question is from the Qing Liu's book: Algebraic Geometry and Arithmetic Curves, Exercise 9.1.6.
Let $X\to S$ be an arithmetic surface and $X_s$ a closed fiber. Let $C_1,...,C_m$ denote the connected components of $X_s$. Let $V\in\operatorname{Div}_s(X)$ be a vertical divisor with support in $X_s$. Show that the following properties are equivalent:
(i) $V\cdot D = 0$ for every $D\in\operatorname{Div}_s(X)$;
(ii) $V^2 = 0$;
(iii) $V\in\bigoplus_{1\le i\le m}\mathbb{Q}C_i$.