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Feb 11, 2017 at 23:34 comment added Jason Starr @ZachTeitler. I re-read the question, and I do not know why you say that. The question is whether existence of simple normal crossings singularities of $X_0$ implies that for an open neighborhood of $t\in S$, the fiber $X_t$ also has simple normal crossings singularities. This does not refer to any ambient variety, but an intrinsic property of singularities.
Feb 11, 2017 at 23:00 comment added Zach Teitler I thought the question wanted X_0 to be SNC in the total space X (not in ambient projective space).
Feb 11, 2017 at 20:40 vote accept Helen
S Feb 11, 2017 at 18:07 history answered Jason Starr CC BY-SA 3.0
S Feb 11, 2017 at 18:07 history made wiki Post Made Community Wiki by Jason Starr