recent paper of \[Xi\], there are ${\bf Q}$-Fano 2-folds and 3-folds (see \[Fl\]) with vanishing $H^0(X, K_X +(-2K_X))$.

Despite of the observations above, in \[Xi\] it is proved that $H^0(X, K_X + (D-K_X)) \ne 0$ for Picard number one Gorenstein del Pezzo surface $X$ and nef and big ${\bf Q}$-Cartier Weil divisor $D$. In this note we shall prove the following which is a consequence of Theorems \[theoremk=0\], \[ruled\], \[theoremk=2\] and \[theoremk=1\] (for the case of integral Cartier $L$, see \[Ka\]).

\[main\] Let $X$ be a nonsingular projective surface. Suppose that $X$ and $L$ satisfy the conditions $(1)$ - $(3)$ in Conjecture \[mainconj\]. Suppose further that $X$ is relatively minimal. Then either $H^0(X, K_X + \lceil L \rceil) \ne 0$ or $H^0(X, K_X + 4L_{\rm red}) \ne 0$.

The second conclusion may occur when $K_X$ is nef (and the Kodaira dimension $\kappa(X) \ge 1$). In this case, the conditions in Conjecture \[mainconj\] are automatically satisfied whenever $L$ is nef and big. So if $L$ is an effective [**Q**]{}-divisor with all coefficients less than $1$, then the non-vanishing of $H^0(X, K_X + \lceil L \rceil)$ is equivalent to that of $H^0(X,
K_X + L_{\rm red})$, which is stronger than our conclusion. 