I had previously asked: Help motivating log-structures
I now have some more questions regarding the role of log structures in moduli problems (you can assume that the moduli problem is the compactification of $n$-marked genus $g$ smooth projective curves for simplicity):
It seems that one of the mantras of the subject is that outside of the boundary, the objects have a unique log-structure. In terms of the example moduli problem I gave, that $n$-marked smooth projective curves of genus $g$ have unique log-structures. In what sense is this true? It doesn't seem literally true to me. Surely they must mean that they have unique log-structures such that they satisfy some property, right? If you can enlighten me about the essence of this mantra, please do!
One of the strengths of log-structures, evidently, is that in the degenerations, they give a unique deformation. So in the example, if we had a stable $n$-marked curve of genus $g$ with a log-structure, there there would be a unique way to extend it to a complete DVR. My question is: what is the virtue of log-structures as opposed to deformation data? Why not instead of a log-structure attached to each (possibly semi-stable) curve, just add some data that will say how it deforms over a complete DVR? Would it be fair to say that log-structures is the natural way to encode this deformation data? Or perhaps there is an extra virtue? I'm confused about this.
Any help would be much appreciated. I've zigzagging between various texts about log-structures, and it is still difficult to get the gist of how to think about them!
P.S. I put this question under community wiki also, but I wasn't sure this time that it was merited. If you have objections, let me know.