Theorem \[0trace-curve\]. In Section 5, we prove Theorem \[0trace-surface\] for the case where $\kappa=1$. In Section 6, we show Theorem \[0trace-surface\] for the case where $\kappa=2$. In Section 7, by using Theorem \[0trace-surface\], we show Theorem \[0threefold-extension\]. In Section 8, we prove Theorem \[0trace-surface-cex\]. In Section 9, we show Theorem \[0surface-ext\].

We summarize literature related to this paper with respect to the basepoint free theorem, the extension theorem, the trace map of Frobenius and the minimal model theory in positive characteristic.

(Basepoint free theorem and extension theorem) The motivation of this paper is the basepoint free theorem and the extension theorem in characteristic zero. Thus, let us summarize some known results on them. Kawamata and Shokurov established the basepoint fee theorem for klt pairs (cf. [@KMM], [@KM]). [@Ambro] generalizes this result (cf. [@Fujino]). The extension theorem is established by Hacon and M^c^Kernan ([@HM Theorem 5.4.21]). This theorem is a key to show the existence of the flip ([@BCHM]). For related topics, see [@DHP] and [@FG].

(The trace map of Frobenius) The heart of this paper is the trace map of Frobenius. The trace map of Frobenius plays a crucial role in the theory of $F$-singularities (cf. [@BST], [@Schwede1], [@Schwede2]). Moreover, 