If we have two lines in $P^3$ which are skewed, then we can take the union of those lines as a subscheme of $P^3$ in order to obtain a subscheme of $P^3$ with a Hilbert Polynomial given by $2m+2$. Moreover, establishing that the coordinates of $P^3$ are $x, y, z$ and $w$, if one of those lines is for example defined by $<x,y>$ and the other line is a member of the 1-parameter family of lines $L_t$ given by $<z,y-tw>$ for $t\neq 0$, then the flat limit of the union of these two skewed lines as $t\to 0$ is a singular conic with an embedded point at $[0:0:0:1]$ whose defining ideal is $<y,xz>\cap<x,y^2,z> $, and that once again gives us a subscheme of $P^3$ with Hilbert Polynomial $2m+2$.
My question is this: Is the given structure for this singular conic $<y,xz>$ in $P^3$ the only one we can take to guarantee Hilbert polynomial 2m+2?
In a similar fashion, the union of three disjoint lines in $P^3$ has Hilbert polynomial $3m+3$, and this configuration of lines by taking an adequate flat family can be specialized to an other configuration with the same Hilbert polynomial, for example, three non planar concurrent lines with an embedded point at the point of concurrence having multiplicity 2 or 3.
But my question is again: if we consider the flat family $X_t$ given by the conic $<z,wx>$ union the line $<w,x+ty>$ with the embedded points $<x,z^2,w>$ and $<x+ty,z,w^2>$ then its flat limit is given by the ideal $<xz,xw,wz>\cap <x^2,z^2,zw,w^2>$ with Hilbert Polynomial 3m+3, is this the only one structure we can take on this singular cubic to guarantee that Hilbert polynomial?
If $X$ is a scheme such that $X_{red} = C$, then $I_X \subset I_C$. Therefore, $X$ is determined by the surjective morphism of coherent sheaves $$ I_C \to I_C/I_X. $$ If $h_X = h_C + 1$ then $I_C/I_X \cong O_x$ for some point $x \in \mathbb{P}^3$. If you want $x$ to be a fixed point of $C$, then the above morphism becomes $I_C \to O_x$ and factors through a surjective morphism $$ I_C/I_C^2 \to O_x. $$ Since $C \subset \mathbb{P}^3$ is a (locally) complete intersection, the conormal sheaf $I_C/I_C^2$ is locally free of rank 2 on $C$, hence there is a $\mathbb{P}^1$ of different maps as above. They give a $\mathbb{P}^1$ of different schemes $X$.