If $π : X → S$ is a family of complex projective varieties, such that $X_0 := π^{−1}(0)$ has simple normal singularities in $X$, then all the general fibers $X_t$ have snc singularities at worst?
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2$\begingroup$ That is not correct. Already in projective $3$-space, a cone over a smooth plane cubic can specialize to a cone over a union of three lines, i.e., a simple normal crossings divisor. $\endgroup$– Jason StarrCommented Feb 11, 2017 at 11:03
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I am just posting my comment as an answer.
That is not correct. Already in projective $3$-space, cones over smooth plane cubics specialize to cones over unions of three lines, i.e., a simple normal crossings divisor consisting of union of three hyperplanes.
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$\begingroup$ I thought the question wanted X_0 to be SNC in the total space X (not in ambient projective space). $\endgroup$ Commented Feb 11, 2017 at 23:00
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$\begingroup$ @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. $\endgroup$ Commented Feb 11, 2017 at 23:34