that the trace map ${{\operatorname{Tr}}}_S^e(A)$ is surjective (resp. a non-zero map). Therefore, to prove the above two theorems, we establish the following results on the trace map of Frobenius.

\[0trace-curve\] Let $k$ be an algebraically closed field of positive characteristic. Let $C$ be a smooth projective curve over $k$. Let $A$ be an ample Cartier divisor on $C$. If $H^0(C, \omega_C(A))\neq 0$, then the trace map $${{\operatorname{Tr}}}_C^e(A):H^0(C, \omega_C(p^eA))\to H^0(C, \omega_C(A))$$ is a non-zero map for every $e\in\mathbb Z_{>0}$.

\[0trace-surface\] Let $k$ be an algebraically closed field of positive characteristic. Let $S$ be a smooth projective surface over $k$. Let $A$ be an ample Cartier divisor on $S$. Assume the following two conditions.

1.  [$K_S+A$ is nef.]{}

2.  [$\kappa(S, K_S+A)\neq 0$.]{}

Then, there exists $m_1\in\mathbb Z_{>0}$ such that the trace map ${{\operatorname{Tr}}}_S^e(A+m(K_S+A))$ $$H^0(S, \omega_S(p^e(A+m(K_S+A))))\to H^0(S, \omega_S(A+m(K_S+A)))$$ is surjective for every $m\geq m_1$ and for every $e\in\mathbb Z_{>0}$.

We would like to consider whether Theorem \[0threefold-extension\] and Theorem \[0trace-surface\] hold for the case where $\kappa(S, K_S+A)=0$. Let us compare Theorem \[0threefold-extension\] with the following basepoint free conjecture (cf. [@KM Theorem 3.3]).

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