$S$ be a smooth prime divisor on $X$ and let $A$ be an ample Cartier divisor on $X$ such that $K_X+S+A$ is nef. Then, $|b(K_X+S+A)|$ is basepoint free for every $b\gg 0$.

If Conjecture \[0bpf\] holds, then Theorem \[0threefold-extension\] also hold for the case where $\kappa(S, K_S+A)=0$. Then, does Theorem \[0trace-surface\] also hold for the case where $\kappa(S, K_S+A)=0$? Unfortunately, the answer is NO. We can construct the following example in characteristic two.

\[0trace-surface-cex\] Let $k$ be an algebraically closed field of characteristic two. Then, there exists a smooth projective surface $S$ over $k$ such that

1.  [$-K_S=:A$ is ample. Note that $\kappa(S, K_S+A)=0$. ]{}

2.  [For every $e\in\mathbb Z_{>0}$, the trace map $${{\operatorname{Tr}}}_S^e(A):H^0(S, \omega_S(2^e A))\to H^0(S, \omega_S(A))$$ is a zero map. ]{}

Moreover, Theorem \[0trace-surface-cex\] also shows that we can not generalize Theorem \[0trace-curve\] to dimension two.

In the appendix of this paper (Section 9), we establish the following analogy of the Hacon–M^c^Kernan extension theorem for surfaces.

\[0surface-ext\] Let $k$ be an algebraically closed field of positive characteristic. Let $X$ be a smooth projective surface over $k$ and let $C$ be a smooth prime divisor on $X$. Let $\Delta:=C+B$ where $B$ is an effective [$\mathbb{Q}$]{}-divisor on $X$ which satisfies 