I am reading Abouzaid's paper "A geometric criterion for generating the Fukaya category" ([https://arxiv.org/abs/1001.4593][1]), and it is claimed there, without proof, in section C.4 in the appendix (pp.34) that the compactificaiton of the moduli space of holomorphic annuli is given by two discs meeting in an interior node at $R\to\infty$ and by two discs meeting at two boundary nodes at $R\to0$, where $e^R$ is the conformal parameter (i.e. annuli is given by $\{1\le|z|\le e^R\}$. I tried searching the literature for a proof, but other than finding the same claim in other papers I have found none.

1. Could someone please suggest a proof, or refer me to one?

2. Is there an explicit way to "calculate" the DM-compactification of a space of Riemann surfaces with boundary (and maybe marked points).?

3. While I can imagine the limit as $R\to\infty$ as a thin neck being developed, I am especially curious about the limit as $R\to 1$ as it seems somewhat arbitraty, why do only 2 chords shrink to a point? why not 1 or 3?

Thank you! [1]: https://arxiv.org/abs/1001.4593