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What are some references for a beginning graduate student in algebraic geometry to learn about the minimal model program? I'm not thinking about entering this field, but rather I just want to know what it means "to run the MMP".

Related link: Training towards research on birational geometry/minimal model program (the OP had a different goal in mind here)

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    $\begingroup$ Matsuki's book "Introduction to the Mori program" is intended, I think, as a more user-friendly alternative to Koll\'ar--Mori. $\endgroup$ Commented Oct 6, 2016 at 18:11

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Of course it depends mostly on your background. But the first chapter, as well as the first half of the second chapter of Kollár-Mori's "Birational Geometry of Algebraic Varieties" is an incredibly inspiring source in my opinion.

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A very light introduction is contained in A first glimpse at the minimal model program. (Also available here.)

I would also second Simone's suggestion: The first two chapters of Kollár-Mori are illuminating.

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