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In the theory of moduli spaces of smooth stable curves with $n$-marked points, I have come across the notion of the coarse underlying curve. Let $C$ be a smooth stable genus $g$ curve with $n$-marked point. We could give it additional structure such as an $r$-spin structure, then there is a natural map to the coarse underlying curve. How is this coarse underlying curve defined and what is this natural map? Is it related to the compactification of the Deligne-Mumford stack or is this something else?

I have not been able to find a definition in any of the papers I am reading. Can someone suggest a reference or give me an intuitive/short description of what this is?

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    $\begingroup$ Where did you find this notion? $\endgroup$
    – abx
    Commented Jul 1, 2019 at 5:51
  • $\begingroup$ @abx See for example page 3 in this paper arxiv.org/pdf/1710.04829.pdf and related references on r-spin closed intersection theory. $\endgroup$
    – user35360
    Commented Jul 1, 2019 at 15:42
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    $\begingroup$ Have a look at the reference [9] in the paper you quote, you'll find a precise definition (in §2.2). $\endgroup$
    – abx
    Commented Jul 1, 2019 at 16:54
  • $\begingroup$ Thank you! I will check it out. $\endgroup$
    – user35360
    Commented Jul 1, 2019 at 18:45

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