Positive cones of $b$-divisor classes

In this article, we define the notion of ample Cartier $b$-divisor classes by using the notion of Seshadri constants for Cartier $b$-divisor classes. In particular, we have shown that the set of all ample Cartier $b$-divisor classes forms a convex cone inside the nef cone of Cartier $b$-divisor classes. Furthermore, we have studied various properties of these Cartier ample $b$-divisor classes. We have also given an equivalent characterization of big Cartier $b$-divisor classes in terms of volume function of the pseudo-effective Cartier $b$-divisor classes. More specifically, we prove that the set of all big Cartier $b$-divisor classes form a convex cone. Finally we have investigated how the nef Cartier $b$-divisor classes behave under the pullback.