Let $X$ be a compact complex manifold and $\beta \in H_2(X, \mathbb{Z}) $ a fixed homology class that is $\textit{decomposable}$. Let $$ \overline{\mathcal{M}}_{0,n}(X, \beta) $$ denote the stable map compactification of the moduli space of degree $\beta$ maps into $X$, with $n$ marked points. Assume that the dimesnion of the moduli space is as extected, i.e. $\delta := <c_1(TX), \beta>-1 +n$.

I have two questions

1) When can one say that an open neighbourhood of any point that belongs to the boundary $\overline{\mathcal{M}} - \mathcal{M}$ is precisely one copy of $\mathbb{C}^{\delta} $. In other words I want to know if the neighbourhood has just one branch or more than one branch. This would be automiatically true if $\overline{\mathcal{M}}$ is non-singular.

2) Let $\overline{\mathcal{M}}_{0,n}(X, \beta; p_1, \ldots, p_k)$ denote the subspace of curves such that the $i^{th}$ marked point goes through $p_i$. When can one conclude that for a generic choice of points $p_1, \ldots,p_k \in X$, an open neighbourhood (inside $\overline{\mathcal{M}}_{0,n}(X, \beta; p_1, \ldots, p_k)$) of a point in the boundary is precisely one copy of $\mathbb{C}^{\delta-k}$.

I particularly want to know if this is true for del-Pezzo surfaces ($\mathbb{P}^1 \times \mathbb{P}^1$ and $\mathbb{P}^2$ blown up at upto $8$ points).

$\textbf{Remark $1$}:$ Another way to phrase the question is, when is the normal neighbourhood of the boundary exactly one copy of $\mathbb{C}$ (i.e. it has exactly one branch).

$\textbf{Remark $2$}:$ I am just wondering if this follows in any way from some gluing map being a surjective map (such as the gluing map described in McDuff and Salamon).

$\textbf{Added Later:} $ I want to clarify; I am asking if the normal neighbourhood of the boundary of the moduli space is $\textit{unibranch}$, not whether it is biholomorphic to an open ball in $\mathbb{C}$.