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I was wondering if there is a generalization of flags in the following way: Suppose you have a series of inclusions of affine varieties $V_1\hookrightarrow V_2\hookrightarrow\cdots\hookrightarrow V_n$ were each inclusion is a closed immersion. Do the set of all such "generalized flags" form a variety? Is this even something that has been studied?

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    $\begingroup$ Yes---look up "flag hilbert scheme." $\endgroup$ Oct 16, 2013 at 6:46
  • $\begingroup$ This reminds me of higher adeles introduced by Beilinson and Parshin. You should be able to get some references by searching these keywords. $\endgroup$ Oct 16, 2013 at 13:28
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    $\begingroup$ I doubt you can get a reasonable scheme in your affine setup. Rather, take everybody projective and fix the Hilbert polynomial of each $V_i$. Then you definitely get a subscheme of the product of a bunch of ordinary Hilbert schemes (which you should see as analogous to, and generalizing, Grassmannians). $\endgroup$ Oct 16, 2013 at 14:44
  • $\begingroup$ @Allen: I completely agree with you, but probably there is also a "flag Haiman-Sturmfels scheme" if one is only interested in affine subschemes of a specific type that are stable with respect to a torus action (i.e., generalizing the notion of affine cones for the usual Hilbert scheme). $\endgroup$ Oct 16, 2013 at 15:51

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