Let $X_0$ be a projective nodal curve. It is known that one can find a smoothing of $X_0$: a family of projective curves $\pi:X\to B$ over a regular curve $B$, which is a smooth morphism over $B\setminus\{b_0\}$, and such that $X_0$ is isomorphic to $X_{b_0}=\pi^{-1}(b_0)$. The surface $X$ can be chosen to be regular.
Question. "How many" smoothings are there for a fixed projective nodal curve $X_0$? Is there a space parametrizing such objects?
I do not see any reasonable moduli functor underlying this problem, but perhaps one can give the family of smoothings of $X_0$ some geometric structure.
If $X_0$ is stable, one can look at an open neighborhood of $[X_0]\in \overline M_g$, which is $(3g-3)$-dimensional. Can we say that by looking in every one (or some) of these $3g-3$ directions we can find a regular smoothing as above?
Thanks for any help.
