# Reference request: unique minimal model in birational class for a complex surface of general type (which may not be smooth)?

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.

• 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. – AG learner Jul 1 at 18:35