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In this article, we prove a finiteness result on the number of log minimal models for 3-folds in
. 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
. 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
of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on
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