Extremal rays of the effective cone

Question

Let $X$ be a smooth projective variety with polyhedral finitely generated effective cone $Eff(X)$. Let $f:X\dashrightarrow X$ be a birational automorphism of $X$ that is an isomorphism in codimension one, that is neither $f$ nor $f^{-1}$ contracts any divisor.

Does $f$ necessarily map a divisor generating an extremal ray of $Eff(X)$ to a divisor with the same property? In other words does the automorphism induced by $f$ on $Pic(X)$ preserve the set of the extremal rays of $Eff(X)$?

Answer 1

Of course: extremal just means that if $D = A+B$ with $A$ and $B$ pseff, then $A$ and $B$ are both proportional to $D$. If $f^*D = A+B$, then $D = (f^{-1})^*(A) + (f^{-1})^*(B)$. If $D$ is extremal, then $(f^{-1})^*(A)$ and $(f^{-1})^*(B)$ are just both $D$ (up to scaling). But then $A$ and $B$ are both proportional to $f*D$, which is extremal as well.