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Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism. Must $f$ necessarily contract a divisor?

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 ...