) = h^0(F,
K_F+\lceil f^*L|_F \rceil ).$$ Since $f^*L$ is big, $f^*L.F >0$. It follows that $\deg(\lceil f^*L|_F \rceil)>0$.

If $g(F) > 0$, then we have $h^0(F, K_F+\lceil f^*L|_F \rceil )
>0$ already. If $g(F)=0$, note that $K_X+L$ is nef. Note also that $(K_{X'}+f^*L).F= (K_X+L).f(F)$ since $F$ is general. This implies that $$\deg(K_F+\lceil f^*L|_F \rceil) =
(K_{X'}+f^*L).F + (\lceil f^*L \rceil -f^*L).F$$ $$=(K_X+L).f(F) + (\lceil f^*L \rceil -f^*L ).F \ge 0.$$ Hence $h^0(F,K_F+\lceil L_F \rceil )>0$. We conclude that ${{\mathcal{F}}}\ne
0$ and hence the required non-vanishing that $h^0(X, K_X+\lceil L
\rceil) \ne 0$.

In the proof of Theorem \[irreg\], without taking log-resolution at the beginning, one can apply Sakai’s lemma [@Sa] for surfaces to get the vanishing of higher cohomology. However, our argument here works for higher dimensional situation. It shows that non-vanishing for general fiber gives the non-vanishing.

Surfaces of Kodaira dimension 0
===============================

In this section, we show that the Conjecture \[mainconj\] in the Introduction is true for surfaces $X$ (not necessarily minimal) with Kodaira dimenion $\kappa(X) = 0$.

\[theoremk=0\] Suppose that $X$ is a nonsingular projective surface (not necessarily minimal) of Kodaira dimension $\kappa(X) = 0$. Then Conjecture \[mainconj\] is true for effective ${\bf Q}$-divisor $L$.

By 