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Here is my question which is a classical result:

Let $f:X\to Z$ be a surjective smooth morphism between smooth projective varieties over an algebraically closed field $k$ of characteristic zero. Let $\dim Z > 0$, then why $-K_{X/Z}$ can not be ample?

In paper [KMM92] Rational connectedness and boundedness of Fano manifolds due to János Kollár, Yoichi Miyaoka and Shigefumi Mori, they state this as a corollary (Corollary 2.8) without proof. In this case $K_X=f^*K_Z+K_{X/Z}$. In [KMM92] they claim that this result can be showed by bend-and-break technique as in Theorem 2.1, but I don't know why? Actually I have no idea how to use bend-and-break technique here???

Thank you for your help!

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  • $\begingroup$ Is $Z$ equal to $Y$? $\endgroup$
    – Matthieu Romagny
    Commented Nov 21, 2023 at 13:33
  • $\begingroup$ @MatthieuRomagny Oh I'm sorry. Now I have edited it. $\endgroup$
    – DVL-WakeUp
    Commented Nov 21, 2023 at 13:35
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    $\begingroup$ If $X \cong Y \times Z$ where $Y$ is a Fano variety and $f$ is the projection then $-K_{X/Z}$ is relatively ample. $\endgroup$
    – Sasha
    Commented Nov 21, 2023 at 14:54
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    $\begingroup$ As KMM state, this follows from bend-and-break. Consider any curve $C$ that is not contained in a fiber, and perform base change by the morphism from $C$ to $Z$. This reduces to the case that $Z$ is itself a smooth, projective curve that admits a rational section of $f$. Now use Bend-and-Break to keep breaking off rational curves in fibers and driving "down" the curve class of the "handle" of the comb. Since there exists some ample divisor class, you cannot do this indefinitely. Once the handle becomes rigid, the relative anticanonical degree is nonpositive. $\endgroup$
    – Jason Starr
    Commented Nov 21, 2023 at 16:21
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    $\begingroup$ Bend-and-break does not only apply to the Fano case. It applies whenever there is a curve whose anticanonical degree is positive. Consider the section curve. If the anticanonical degree is positive, use Bend-and-Break to "break" off a rational curve in a fiber so that the original section curve is algebraically equivalent to the broken-off rational curves union a "handle" that is also a section curve, but now with smaller anticanonical degree. If the anticanonical degree is still positive for this handle, repeat. $\endgroup$
    – Jason Starr
    Commented Nov 22, 2023 at 10:10

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