I'm currently reading about the minimal model program for varieties (defined over $\mathbb{C}$). I've studied the fundamental results (cone theorem, kawamata base point free theorem and contraction theorem), and I've understood the general idea of the programs through the flow-char of the contractions. Unfortunately, I haven't read any specific example yet, but still I have some heuristic question about MMP (I hope they're not silly).
- in practice, does it matter for instance the choice of the $K_X$-negative ray you start with? Could it happen that a "bad" choice of a ray leads you to a variety whose canonical divisor is not nef, but still you can't contract curves?
- could you exchange the order of contractions (assuming they still make sense, because there are different varieties in play)?
I've tried to look by myself but wasn't able to find a reference, any help woule be much appreciated, thanks in advance!