Say we have a flat family $ \{X_t \}_{t \in \mathbb{P}^1-{0}}$ with $X_i$ sub-schemes of $\mathbb{P}^n$. Assume that each $X_i$ does not lie in a hyerplane. Is it true that the limit $X_0$ also doesn't lie in a hyperplane? I want to use some kind of semi-continuity argument but I am unable to find the right cohomology group. If this isn't true in general, under what conditions is it true.
The motivating example is of course, the twisted cubic in $\mathbb{P}^3$ degenerating to a plane cubic with an embedded point. In this case the the embedded point points out of the plane and so the limit doesn't lie in a hyperplane.