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Nov 12, 2015 at 19:48 comment added ChrisLazda Ah that's actually really useful. I guess I had just naively assumed that you could glue the above log structure to get something on the whole of $X_k$, but clearly this was a mistake.
Nov 4, 2015 at 3:21 comment added Mattia Talpo I think that your charts on the central fiber are the correct ones, but it is not clear (without knowing that there is a smoothing) that they define a "global" log structure. Sometimes you can't put log structures of that type on normal crossing varieties. The discussion in section 5 here arxiv.org/pdf/1006.5870v1.pdf should be relevant.
Nov 2, 2015 at 17:12 history asked ChrisLazda CC BY-SA 3.0