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  • here is an example where $B_1$ is not a finite union of cones: Let $X$ be a K3 surface of Picard number 3, such that the cone of effective divisors, $Eff(X)=Nef(X)$ is one of the components of the $[D\in NS(X) | D^2\ge 0]$ (for example, a K3 surface without $(-2)$-curves, such surfaces can be constructed as in this paper).
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