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Remark 10.1.8 in Liu's book (AG and Arithmetic curves) says that over a non-affine base (base is always assumed to be a Dedekind scheme of dim 1), the minimal regular model of a (smooth projective) curve might not exist. There, it's said that over a non-affine base, we might get something proper, but not projective (projectivity is required in their definition of regular minimal model).

This is where I am a bit confused. What I have in mind is as follows. I start with any integral projective model. Resolve singularity by normalization and blowing-ups, which preserves projectivity. Then blow down all the vertical (-1)-curve, using Castelnuovo, which also preserves projectivity. This will give me a relatively minimal regular model, which is the same as a minimal model when the curve I start with has genus >= 1.

I hope someone can point out where the problem is.

A related question is why we really need projectivity? Is there anything that we can't do with just local projectivity?

Thank you.

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If the base is not affine then there are several inequivalent definitions for "projective over the base" (e.g., definitions of "projective morphism" in Liu's book and in EGA are not equivalent), but it is odd to demand projectivity in any of these senses since Zariski-locally on the base it is automatic in every reasonable sense. It is not a standard convention in the literature to demand a projectivity condition in the definition of the minimal regular proper model of a curve over a Dedekind base, but could be a convenience for the use of fibered surfaces in Liu's book. – ayanta Dec 25 at 4:53
Dear ayanta, thanks for your comment. So in the process I described above, are you saying that blow-up is projective in the sense of EGA but not in the sense of Liu's book? – QcH Dec 25 at 16:16
Dear QcH: Yes (and the definition of "projective morphism" in Liu's book is the same as in Hartshorne's textbook on algebraic geometry; the latter book notes the disagreement with EGA near the end of the Introduction). – ayanta Dec 25 at 18:09
Thank you. That clarifies many things. I didn't notice that Liu's book also uses this more restrictive version of projectivity. – QcH Dec 25 at 19:47

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