I refer to the paper "Normal Subgroups in the Cremona Group". In the last paragraph of the proof of proposition 5.13, the author wrote the following: "Assume now that $h\in \text{Bir}(X)$ preserves the axis of $g_\ast$. Then $h_\ast [D']$ is an ample class." Here $[D']$ is an ample class in $N^1(X)\cap V_g$. May I ask how is the ampleness of $h_\ast[D']$ deduced?


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