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Is there an analogue of Knudsen clutching for moduli stacks of "pointed" varieties?

I admit this question is not very precise. I'm really asking two questions though.

Are there analogues of the moduli stack of (g,n) curves in higher dimension, where we consider say varieties with hilbert polynomial h and n "points" where I don't know yet what I mean by points precisely?

Do these moduli stacks, if they exist at all, have some kind of Knudsen clutching?

Still not precise, but hopefully a bit more clear.

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  • $\begingroup$ This question makes no sense as stated. Precisely what moduli stacks of pointed varieties are you considering? $\endgroup$ May 4, 2013 at 9:25
  • $\begingroup$ I'd primarily like to consider the moduli stacks $M_h$ of can. polzd varieties, where h is some hilbert polynomial. Then I wonder whether one can define in a sensible way the analogues of $M_{g,n}$ in this higher dimensional setting, and do some Knudsen clutching. $\endgroup$
    – Jonathan
    May 4, 2013 at 9:41
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    $\begingroup$ BTW, what is Knudsen clutching? this may help $\endgroup$
    – IMeasy
    May 4, 2013 at 12:30

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