the following properties.

1.  [$C\not\subset{{\operatorname{Supp}}}B$, $\llcorner B\lrcorner=0$ and $(X, \Delta)$ is plt. ]{}

2.  [$B\sim_{\mathbb Q}A+F$ where $A$ is an ample [$\mathbb{Q}$]{}-divisor and $F$ is an effective [$\mathbb{Q}$]{}-divisor such that $C\not\subset {{\operatorname{Supp}}}F.$]{}

3.  [No prime component of $\Delta$ is contained in the stable base locus of $K_X+\Delta.$]{}

Then, there exists an integer $m_0>0$ such that, for every integer $m>0$, the restriction map $$H^0(X, mm_0(K_X+\Delta))\to H^0(C, mm_0(K_X+\Delta)|_C)$$ is surjective.

But, the proof for Theorem \[0surface-ext\] does not use the trace map of Frobenius. We use results on the minimal model theory established in [@T2] and [@T3].

In Section 1, we summarize the notations. In Section 2, we see the definition and some basic properties of the trace map of Frobenius. The trace map of Frobenius is obtained by applying the functor $\mathcal Hom_{\mathcal O_X}(-, \omega_X)$ to the Frobenius map $\mathcal O_X\to F_*\mathcal O_X.$ In Section 3, we see some known facts on Cartier operator. We can consider the trace map of Frobenius as Cartier operator. The Cartier operator is defined by the de Rham complex. We use Cartier operator to consider the relation between the trace map of Frobenius and étale base changes. In Section 4, we show Theorem \[0surface-special-extension\] and 