[@CHMS] and [@Mustata] establish results related to birational geometry by using the trace map of Frobenius and the theory of $F$-singularities. For related topics, see [@BSTZ] and [@TW].

(Minimal model theory in positive characteristic) Let us summarize literature on the minimal model theory in positive characteristic. For results on surfaces, see [@Fujita3], [@KK], [@T2] and [@T3]. For results on threefolds, see [@Kawamata], [@Keel], [@Kollar] and [@HX].

The author would like to thank Professor Osamu Fujino for many comments and discussions. He would like to thank Professor Karl Schwede for numerous helpful comments. He thanks Professor Atsushi Moriwaki for warm encouragement. The author is partially supported by JSPS Fellowships for Young Scientists.

Notations
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We will freely use the notation and terminology in [@KM].

We will not distinguish the notations invertible sheaves and divisors. For example, we will write $L+M$ for invertible sheaves $L$ and $M$.

For a coherent sheaf $F$ and a Cartier divisor $L$, we define $F(L):=F\otimes \mathcal O_X(L)$.

Throughout this paper, we work over an algebraically closed field $k$ of positive characteristic and let ${\rm char}\,k=:p>0.$

In this paper, [*a variety*]{} means an integral scheme which is separated and of finite type over $k$.

The trace map of 