equal to $0$. Pick up a general fibre $F$ of $f$. The induction of the non-vanishing problem on $F$ may imply that $$\text{rk}(f_*(\omega_{V/B}\otimes {\mathcal
O}_V(\lceil L\rceil)))=h^0(F, K_F+\lceil L\rceil|_F)\ge h^0(F,
K_F+\lceil L|_F\rceil)\ne 0.$$

The positivity of $f_*(\omega_{V/B}\otimes {\mathcal O}_V(\lceil
L\rceil))$ has direct applications in studying properties of the moduli schemes for polarized manifolds. Please refer to [@Vi2] for more details.

The above remark shows one aspect of the importance of the effective non-vanishing for [**Q**]{}-divisors.

We would like to thank the referee for his / her very careful reading and suggestions for the improvement of the paper.

Some preparations and an example {#prep}
================================

We begin with:

A reduced connected divisor $\Gamma$, with only simple normal crossings, is a rational tree if every component of $\Gamma$ is a rational curve and the dual graph of $\Gamma$ is a tree (i.e., it contains no loops).

Before proving Proposition \[treeprop\] below, we need two lemmas in advance.

\[rat\] Let $D = \sum_{j=1}^n D_j$ be a reduced connected divisor on a nonsingular projective surface $X$. Then $D . (K_X + D) \ge -2$ and the equality holds if and only if $D$ is a rational tree.

Note that $\sum_{k < j} D_k . D_j 