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I am not sure if this question is appropriate for this site, but here it goes. I am not a geometer, so I am not familiar with notation in the area.

I am interested in the moduli space of $r$-spin structures. Let $[C,p_1,...,p_n] ∈ \mathcal{M}_{g,n}$ be a nonsingular curve with distinct marked points. Then some authors define a log-canonical bundle, denoted by $\omega_C(\sum p_i)$ (for example this paper)? What is a log-canonical bundle as opposed to a canonical bundle and what does the data $(\sum_i a_i p_i)$ where $a_i$s are integers meant in this context?

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  • $\begingroup$ Are you asking for the philosophy behind the introduction of these line bundles, or just for their definition? In the latter case the question is not appropriate for this site, you'll find the definition in any book on algebraic curves. $\endgroup$
    – abx
    Commented Jun 10, 2019 at 4:05
  • $\begingroup$ @abx I am asking for both of those things. I couldn't find a definition either. Please could you give me a reference? $\endgroup$
    – user35360
    Commented Jun 10, 2019 at 4:21
  • $\begingroup$ Arbarello-Cornalba-Griffiths-Harris, Geometry of algebraic curves. $\endgroup$
    – abx
    Commented Jun 10, 2019 at 4:43
  • $\begingroup$ also ' lectures on vanishing theorems' is way more general but contains enough theorems to explain why log structures are important. $\endgroup$
    – meh
    Commented Jun 10, 2019 at 18:05

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