  In this article we prove a finiteness result on the number of log minimal
models for $3$-folds in char $p>5$. We then use this result to prove a version
of Batyrev's conjecture on the structure of nef cone of curves on $3$-folds in
characteristic $p>5$. We also give a proof of the same conjecture in full
generality in characteristic $0$. We further verify that the duality of movable
curves and pseudo-effective divisors hold in arbitrary characteristic. We then
give a criterion for the pseudo-effectiveness of the canonical divisor $K_X$ of
a smooth projective variety in arbitrary characteristic in terms of the
existence of a family of rational curves on $X$.
