While studying and solving some exercises on Hilbert schemes, I've come across many problems in Hartshorne's book on deformation theory which ask the reader to show certain properties such as irreducibility, openness or non-singularity. Below I list a few examples.
(Assume $ k $ is an algebraically closed field.)
(1) In $ \operatorname{Hilb}^{3t+1} ( \mathbb{P}^3_k ) $, the twisted cubic curves form a non-singular open subset of an irreducible component $ H_0 $ of dimension $ 12 $, also courtesy of Piene and Schlessinger's paper On the Hilbert scheme compactification of the space of twisted cubics. This Hilbert scheme has another irreducible component $ H'_0 $ corresponding to plane elliptic curves union a point not on the curve.
(2) In $ \operatorname{Hilb}^8( \mathbb{P}^4_k ) $, there is an irreducible component of dimension $ 32 $ that has a non-singular open subset corresponding to $ 8 $-tuples of distinct points in $ \mathbb{P}^4_k $.
(3) The complete intersection curves in $ \mathbb{P}^3_k $ obtained by homogeneous polynomials of degrees $ a,b $ form a non-singular open subset of an irreducible component of the corresponding Hilbert scheme.
Question: How does one justify these properties, particularly the openness of certain subsets and irreducibility of components? Although these assertions seem intuitively clear, are there strategies in general that allow you to see these rigorously? For example, in the paper by Piene and Schlessinger, $ H'_0 $ is stated to be irreducible without proof and I'm not sure how to do this.
