Relative Cartier divisor scheme

Question

This is question 1.14.1.2 in Kollar's book Rational Curves on Algebraic Varieties. Let $f:X\rightarrow S$ be a flat and projective morphism with integral fibres such that $H^1(X_s,O_{X_S})=0$ for every $s\in S$. Assume that $S$ is reduced and connected. Let $L$ be a line bundle on $X$ such that $h^0(X_s,L_s)$ is independent of $s\in S$. Then obviously $f_*L$ is locally free and $Proj_S(f_*L)$ is embedded in $CDiv(X/S)$. However, I cannot show that $Proj_S(f_*L)$ is a connected component. Here $CDiv(X/S)$ is the representable scheme of relative effective Cartier divisor: $$CDiv(X/S)(Z)=\{\mathrm{Relative\; effective\; Cartier\; divisors\; of\; }V\subset X\times_S Z\}.$$