Algebraic geometry – Flat limit (of twisted cubes) contained in surfaces

Let $ H $ denotes the irreducible component of $ text {Hilb} ^ {3t + 1} mathbb {P} ^ $ 3 whose general member corresponds to a non-singular twisted cubic. Let $ C $ be a sub-regime within the limit of $ H $ and suppose that it is in a surface $ S subseteq mathbb {P} ^ $ 3.

So why is it possible to find families $ C_R, S_R subseteq mathbb {P} ^ 3_R $ on a DVR $ R $ with fraction field K $ such as

1) $ C_R subseteq S_R $

2) Generic fiber $ C_K $ is a non singular twisted cubic

3) $ C subseteq S $ are the closed fibers of the family.

The authors in "Hilbert Diagram Compactification of Twisted Cubics Space": https://www.uio.no/studier/emner/matnat/math/MAT4230/h10/undervisningsmateriale/Hilbertscheme.pdf on display at the page 4 (pg 763), line 7 of the evidence. Although they are studying built-in points, this statement seems to be something more general about uniform boundaries?

More generally, is it true that if something on the boundary of my component in a Hilbert schema was located in a hypersurface, could I then find a family on a DVR as above?