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Is there any nice description/picture of the moduli space of stable disks with 1 interior marked point and 4 marked points on the boundary?

enter image description here

I'm expecting it to be a 3-dimensional polytope, because if we fix the interior point and, say the green marked point, I have 3-degree of freedom for moving the remaining marked point.

Is it possible to know, say, the number of edges each faces should have? the number of vertices and so on?

I was wondering if there are some references where this space is already studied, before trying to reinvent the wheel.

If there are any trick for finding all the limiting configurations I'm very interested in knowing them as well.

Thanks in advance.

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A generalization of this moduli space was used by Costello, in the paper https://arxiv.org/abs/math/0601130. In his terminology, your space is $D_{0,1,4,1} = \overline{\mathcal N}_{0,1,4,1}$.

The combinatorics of the boundary strata are described by isomorphism classes of genus $0$ stable ribbon graphs with $4$ labelled external edges, and one labelled internal vertex. And Lemma 2.2.4 describes each stratum as an orbicell.

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  • $\begingroup$ Thanks for pointing out that paper! In figure 1 the author draw a picture of an element of $D_{0,2,4,2}$ (i.e. annuli with 4 marked points on the boundary and 1 interior) as three disk glued together at three nodes. Do you happen to know why? moreover that configuration is not a tree, and in a Mumford compactification of the moduli space we only have "tree"- like bubbling phenomena $\endgroup$
    – Riccardo
    Apr 29, 2020 at 1:23
  • $\begingroup$ I can't say much beyond the definitions in the paper. The figure is a degenerate configuration where the annuli is "squeezed". Only $\overline N_{0,2,4,2}$ will contain actual annuli. In your case-- the case of $D_{0,1,4,1}$-- only trees will appear. $\endgroup$
    – Phil Tosteson
    Apr 29, 2020 at 1:52

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