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.