that $K_X+S+A$ is nef. Fix $m\in\mathbb Z_{>0}.$ Then, by the Kodaira vanishing theorem, we obtain $$H^1(X, K_X+A+(m-1)(K_X+S+A))=0.$$ Thus, the natural restriction map $$H^0(X, m(K_X+S+A))\to H^0(S, m(K_S+A))$$ is surjective.

It is natural to consider whether the above fact also holds in positive characteristic. Unfortunately, however, there exists the following example.

Let $k$ be an algebraically closed field of positive characteristic. Then, there exist a smooth projective surface $X$ over $k$, a smooth prime divisor $C$ on $X$ and an ample Cartier divisor $A$ on $X$ such that $K_X+C+A$ is nef and that the natural restriction map $$H^0(X, K_X+C+A)\to H^0(C, K_C+A)$$ is not surjective.

Thus, we would like to find a suitable analogy of Fact \[0char-zero\] in positive characteristic. In this paper, we prove the following two theorems.

\[0surface-special-extension\] Let $k$ be an algebraically closed field of positive characteristic. Let $X$ be a smooth projective surface over $k$. Let $C$ be a smooth prime divisor on $X$ and let $A$ be an ample Cartier divisor on $X$. If $H^0(C, K_C+A)\neq 0$, then the natural restriction map $$H^0(X, K_X+C+A)\to H^0(C, K_C+A)$$ is a non-zero map.

\[0threefold-extension\] Let $k$ be an algebraically closed field of positive characteristic. Let $X$ be a smooth projective threefold 