TITLE: When can stable map space have non-reduced structure?
QUESTION [6 upvotes]: My question is on which situation stable map space $\overline{M_{g,n}}(X,\beta)$
can have non-reduced structure.
There is many example of stack structure from automorphism of stable curve
but I don't know any example of non-reduced structure
And more explicitly, when (g,n) = (0,0), is it still posible that
$\overline{M_{0,0}}(X,\beta)$ can have non-reduced stack structure?

REPLY [3 votes]: There are examples of the following form. Let $C\subset X$ be a contractible curve in a threefold which is isomorphic to $\mathbb{P}^1$ and has normal bundle $\mathcal{O}\oplus\mathcal{O}(-2)$. Such examples were studied by Miles Reid and have very explicit local models. In such a situation, the inclusion $C\to X$ is an isolated point in $\overline{M}_0(X,[C])$ which has non-reduced structure. The infinitesimal deformation given by the non-zero section of $N_{C/X}$ is obstructed. The length of the non-reduced structure is an invariant that Mile Reid calls width. 
Explicitly, let $V$ be the affine hypersurface in $\mathbb{C}^4$ given by $x^2+y^2+z^2+w^{2k}=0$ where $k\geq 2$. Let $X\to V$ be the small resolution of this singularity. The exceptional set of this resolution is $C$. Then $\overline{M}_0(X,[C])\cong Spec(\mathbb{C}[t]/(t^k))$ .