Why does the MMP terminates when the canonical divisor is nef?

I have recently taken interest in Mori's Minimal Model Program (MMP) and I struggle to figure out why it stops when the canonical divisor $$K_X$$ of our variety $$X$$ is nef.

For now, I have understood the following:

• If $$X$$ is a normal and projective variety with terminal singularities whose canonical divisor $$K_X$$ is not nef, then the Cone Theorem tells us that there are rational curves $$C$$ that are $$K_X$$-negative and that generate an extremal ray in the closed cone of curves $$\overline{NE}(X)$$. Moreover, if $$g: X \rightarrow X'$$ is a divisorial contraction of such extremal ray, then $$X'$$ is also a normal and projective variety with terminal singularities.

• If $$X$$ is moreover nonsingular and $$2$$-dimensional, then the contraction $$g:X \rightarrow X'$$ of a $$K_X$$-negative extremal ray induces a variety $$X'$$ that is nonsingular as well.

• If $$Y$$ is a (smooth projective) $$2$$-dimensional variety whose canonical divisor $$K_Y$$ is nef, then (via the Adjunction Formula) $$Y$$ has no $$(-1)$$-curves, and therefore (via Castelnuovo's Theorem) there is no birational morphism $$\mu: Y \rightarrow Y'$$ (where $$Y' \neq Y$$ is a smooth projective surface). A (smooth projective) surface with nef canonical divisor is therefore minimal.

My question is the following: In the general case, if $$Y$$ is a minimal model (i.e. a projective variety with terminal singularities and whose canonical divisor $$K_Y$$ is nef),then does that mean that there is no birational morphism $$\mu: Y \rightarrow Y'$$(where $$Y' \neq Y$$ is a projective variety $$Y'$$ with terminal singularities)?

In general, I think the most we can say is that if $$\mu: Y \to Y'$$ is such a morphism, then $$\mu$$ is small; it has no exceptional divisors.

In this situation, $$K_{Y/Y'} := K_Y - \mu^* K_{Y'} = \sum a_E E$$ where this sum ranges over all the exceptional divisors of $$\mu$$. Moreover $$a_E > 0$$ for all $$E$$, since $$Y'$$ has terminal singularities. This divisor is $$\mu$$-nef because if $$C \subset Y$$ is a curve contracted by $$\mu$$, then $$(K_{Y/Y'}).C = K_Y.C \geq 0$$, since $$Y$$ is a minimal model.

By the negativity lemma, we know $$- \sum a_E E$$ is effective, which can only be the case if $$\mu$$ has no exceptional divisors.

To be honest, I'm having trouble finding examples of such small morphisms between minimal models but I strongly suspect they exist in general. (If I find one which is easy to describe I'll edit it in.)

However, there are some settings where what you say holds.

For example, suppose there is a curve $$C \subset Y$$ contracted by $$\mu$$ so that $$K_Y.C > 0$$. Then, $$K_{Y'}$$ is not $$\mathbb{Q}$$-Cartier. Indeed, if $$m > 0$$ is so that $$m K_Y$$ and $$mK_{Y'}$$ are Cartier, then $$\mu^* m K_{Y'}$$ and $$m K_Y$$ agree outside of a codimension $$\geq 2$$ set, which implies they are linearly equivalent. This is absurd since $$(\mu^* m K_{Y'}).C = 0$$.

As such, if $$K_Y$$ is ample then the exceptional set of any such $$\mu$$ would have to be empty, so $$\mu$$ would have to be an isomorphism.

• I apologize for the edit and unedit. I thought I came across an example today but I was mistaken: the example did not have positive canonical bundle. Commented Jun 2 at 17:19