# 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?