# When is the normal neighbourhood of the boundary of the moduli space of cuvres parametrized by exactly one branch?

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 $\mathbb{C}$.

• I want to clarify: are you asking whether or not the moduli space is locally smooth, i.e., locally biholomorphic to a complex ball? Or are you asking whether the moduli space is unibranch? These are quite different things. At any rate, the $(-1)$-curves on del Pezzo surfaces will cause most moduli spaces to be reducible, so they will be neither smooth nor unibranch. – Jason Starr Jun 18 '15 at 10:00
• @Jason: I am asking whether it is unibranch. – Ritwik Jun 18 '15 at 11:55
• @Jason: I think I am tacitly assuming that the curve whose normal neighbourhood I am interested in is a decomposable curve. For del-Pezzo surfaces the question I really want to ask is if you have a decomposable curve in the boundary, is its normal neighbourhood unibranch? – Ritwik Jun 18 '15 at 12:12

I am not completely certain if this is what Ritwik is asking, but there are points in the boundary of the moduli space where the moduli space is reducible. Let $A$ and $B$ be copies of $\mathbb{P}^1$. Form the product $A\times B$, i.e., $\mathbb{P}^1\times \mathbb{P}^1$, a smooth quadric surface, the Hirzebruch surface $\mathbb{F}_0 = \Sigma_0$, etc. Let $\nu:X\to A\times B$ be the blowing up at a point $(a,b)$. Let $E$ denote the exceptional divisor. Let $\widetilde{B}$ denote the strict transform of $\{a\}\times B$. Let $\beta$ denote the curve class $2([E] + [\widetilde{B}])$.
Every stable map parameterized by the "interior" $\mathcal{M}_{0,0}(X,\beta)$ is a double cover of $\{a'\}\times B$ for some $a'\in A\setminus\{a\}$. In particular, the "interior" is smooth and connected of dimension $\delta = 3$. There is one parameter for $a'$ in the curve $A\setminus\{a\}$, and there are two parameters for the two branch points / ramification points.
However, the space of all stable maps $\overline{\mathcal{M}}_{0,0}(X,\beta)$ also parameterizes reducible stable maps $u:C\cup D \to \widetilde{B}\cup E$ such that $C$ is a double cover of $\widetilde{B}$, resp. $D$ is a double cover of $E$. The boundary locus $\Delta_{2[\widetilde{B}],2[E]}$ parameterizing such stable maps has dimension $\delta + 1 = 4$: two parameters for the ramification points of $u|_C$ and two parameters for the ramification points of $u|_D$. Such a stable map is in the closure of $\mathcal{M}_{0,0}(X,\beta)$ if and only if the intersection point of $C$ and $D$ is a ramification point both of $u|_C$ and of $u|_D$. If you consider the moduli space in a neighborhood of such a point, it has two irreducible components.