a nonsingular projective surface $X$ and a [**Q**]{}-divisor $L$ such that the conditions (1) and (2) in Conjecture \[mainconj\] are satisfied, but that $H^0(X, K_X + \lceil L
\rceil) = 0$. Indeed, we will see that both $L$ and $K_X + L$ are nef and big ${\bold Q}$-divisors.

Let $C$ be a sextic plane curve with $9$ ordinary cusps (of type $(2, 3)$) and no other singularities. This $C$ (regarded as a curve in the dual plane ${\bf P}^{2*}$) is dual to a smooth plane cubic (always having $9$ inflectins). Let $\overline{X}
\rightarrow {\bf P}^2$ be the double cover branched at $C$. Then $\overline{X}$ is a normal $K3$ surface with exactly $9$ Du Val singularities (lying over the $9$ cusps) of Dynkin type $A_2$. Let $X$ be the minimal resolution. According to Barth \[Ba\], these $9
A_2$ are $3$-divisible. That is, for some integral divisor $G$, we have $3 G \sim \sum_{i=1}^9 (C_i + 2D_i)$ where $\coprod (C_i +
D_i)$ is a disjoint union of the $9$ intersecting ${\bf P}^1$ (i.e., the $9A_2$). Let $H$ be the pull back of a general line away from the $9$ cusps on $C$. Then $H^2 = 2$ and $H$ is disjoint from the 