# Moving curves and small transformations

Let $f:X\dashrightarrow Y$ be an isomorphism in codimension one between smooth projective varieties. Let $C\subset X$ a curve generating an extremal ray of the cone of moving curves $Mov_1(X)$, and let $\Gamma \subset Y$ be its strict transform via $f$.

Does $\Gamma$ generate an extremal ray of $Mov_1(Y)$ ?

• To reach a wider audience, you might define more of your terms, e.g., moving curves $Mov_1(X)$. – Joseph O'Rourke Jan 27 '17 at 23:04
• In characteristic $0$ that follows immediately from the big work of Boucksom-Demailly-Paun-Peternell: $f$ preserves the cone of effective divisors, and thus also the closure of this cone. The closure of this cone is dual to the cone of moving curves (or perhaps its closure ... maybe that is what you are asking about). At any rate, you do not need that to deduce the result: it should follow in all characteristics from the projection formula. – Jason Starr Jan 28 '17 at 10:53