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Log structures, semistable degenerations, log crystalline cohomology, log de Rham cohomology, log smoothness, log Gromov-Witten theory
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A log structure on the moduli space of curves
(I'm not sure this is a real answer, but it was too long for a comment)
I've never seen the log structure you consider.
By "log moduli space of (stable) log curves" usually one refers to the natural …