I don't seem to have the option to edit my answer, but I wanted to correct myself: it is the scheme structure of the limit that determines the possible nearby varieties in the family, not vice versa. The question of whether the nearby varieties determine the limit is that of Hausdorfness of the parameter space. One has to restrict the nature of the possible limits to get uniqueness of the limit. This arises in deciding just how complicated the limit should be when trying to construct a nice compactification of a given family of nice varieties. It is a wonderful fact that for smooth curves one can compactify them without introducing non varieties. All one needs is some simple singular curves. This was first noticed by Alan Mayer and David Mumford in their talks at the Woods Hole conference 1964, whose notes appear on James Milne's webpage at Michigan, (and mine at UGA). I am roy smith, and have been using mathwonk as alias for so many years online I have forgotten it is not my name. Is it sufficient to register and add it to my profile, or do I need to sign posts here as roy smith? That would apparently unlink me from all my mathwonk activity so far.
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I don't seem to have the option to edit my answer, but I wanted to correct myself: it is the scheme structure of the limit that determines the possible nearby varieties in the family, not vice versa. The question of whether the nearby varieties determine the limit is that of Hausdorfness of the parameter space. One has to restrict the nature of the possible limits to get uniqueness of the limit. This arises in deciding just how complicated the limit should be when trying to construct a nice compactification of a given family of nice varieties. It is a wonderful fact that for smooth curves one can compactify them without introducing non varieties. All one needs is some simple singular curves. This was first noticed by Alan Mayer and David Mumford in their talks at the Woods Hole conference 1964, whose notes appear on James Milne's webpage at Michigan, (and mine at UGA). |
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