Projective manifold whose anticanonical section is composed of two components

Question

Let $M$ be a connected projective complex manifold with a smooth anticanonical divisor $D$ ($D \sim -K_M$). In an answer to a previous question, It is told that $D$ may have at most two components.

An easy example is $M= \mathbb P^1 \times X$, where $X$ is a projective manifold with trivial canonical class.

Any blow-up of $M$ along a smooth variety that is a smooth divisor of $D$ is again such an example.

Here is my question:

Is a projective manifold whose anticanonical section is composed of two components always birational to $\mathbb P^1 \times X$, where $X$ is a projective manifold with trivial canonical class?

Of course, it is true for two dimension. I am curious about higher dimensional cases.

Answer 1

If you look at the proof of the theorem referenced in the answer to that linked question, near the bottom of p.801 and top of p.802 it is established that if you run an MMP (the chosen boundary divisors specified in the proof), the first time you encounter a Fano contraction (which you must), then it is a $\mathbb P^1$-bundle. In fact, this is how the fact that there can be at most two components is established. As the previous steps of the MMP are birational, this means that the original variety is birational to this $\mathbb P^1$-bundle.