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Frankly I know nothing about this type of algebraic geometry (or any kind of graduate level AG), but I need at one point in an article to say "there exists a unique minimal model, since the surface is of general type", and I can't seem to find a reference for this statement.

I also have a hunch that for some surfaces, being minimal is equivalent to having no (-1) curves? It would be nice to able to state it in this fashion, since I understand what that means...!

Thank you in advance for any help.

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  • $\begingroup$ In general, if a smooth compact complex surface has Kodaira dimenison $\ge 0$, then the minimal model is unique, see Barth-Hulek-Peteres-Van de ven's Compact Complex Surfaces, chapter III, prop. 4.6. Besides, being minimal is the same as no $(-1)$ curve is right. $\endgroup$ – AG learner Jul 1 at 18:35
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Such a reference can be located by searching for the correct words on your favorite search engine and a little persistence. For instance, starting with "minimal models for surfaces," one arrives at the Wikipedia page for the minimal model program, reads down to the section on minimal models of surfaces, which links to the page on the Enriques-Kodaira classification, and in the references on this page there are a great many sources. One which is freely available on the internet is Chapters on Algebraic Surfaces, by Miles Reid, which contains the result you ask for in section C.3 on page 57.

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