I am interested in the following situation. If $S_1\cup_D S_2$ is a union of two irreducible smooth projective surfaces over $k=\overline{k}$(over $k=\mathbb{C}$ is enough, if it's relevant) glued along a smooth divisor $D$, when can we realize it as a degeneration of a smooth projective surface $S$?
To be precise, I want to know when we can find a proper flat morphism $\pi:\mathcal{S}\to C$ where $C$ is a smooth projective curve and $\pi^{-1}(t)$ is deformation equivalent to the surface $S$ for $t\neq0\in C$ but over $0\in C$, we have $S_1\cup_C S_2$.
The case that I am primarily interested in is the case of $S_1$ and $S_2$ both being rational surfaces. For instance, by taking $xy=t$ in $\mathbb{P}^3$, we can see that a quadric surface can be degenerated to a union of two copies of $\mathbb{P}^2$ glued along a line in $\mathbb{P}^2$.
The other example that I know of is to use degeneration to the normal cone, starting from a smooth divisor $D\subset S$, this gives a degeneration to $S_1\cup_DS_2$, $S_1=S$ and $S_2$ being a ruled surface. However, this construction cannot reproduce reducible surfaces, such as $\mathbb{P}^2\cup\mathbb{F}_3$, where we glue a line to a unique $(-3)$ section in the Hirzebruch surface $\mathbb{F}_3$ or $\mathbb{P}^2\cup\mathbb{F}_6$ along conics in $\mathbb{P}^2$ and $(-6)$ section in $\mathbb{F}_6$.
Question. Is there any existence result on the degeneration of a single surface(given a reduced reducible surface as the central fiber) or is there any other construction of such degeneration other than examples above in the literature?
