Minimal model theorem for toric divisors

Minimal model conjecture for a proper variety $X$ is that if $\kappa(X)\geq 0$, then $X$ has a minimal model with the abundance and if $\kappa =-\infty$, then $X$ is birationally equivalent to a variety $Y$ which has a fibration $Y \to Z$ with $-K_Y$ relatively ample. In this paper, we prove this conjecture for a $\D$-regular divisor on a proper toric variety by means of successive contractions of extremal rays and flips of ambient toric variety. Furthermore, for such a divisor $X$ with $\kappa(X)\geq 0$ we construct a projective minimal model with the abundance in a different way; by means of "puffing up" of the polytope, which gives an algorithm of a construction of a minimal model.