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 $\mathbb{C}$.

  • $\begingroup$ 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. $\endgroup$ Commented Jun 18, 2015 at 10:00
  • $\begingroup$ @Jason: I am asking whether it is unibranch. $\endgroup$
    – Ritwik
    Commented Jun 18, 2015 at 11:55
  • $\begingroup$ @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? $\endgroup$
    – Ritwik
    Commented Jun 18, 2015 at 12:12

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.