When can stable map space have non-reduced structure? 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?
 A: 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 $\overline{M}_0(X,[C])\cong Spec(\mathbb{C}[t]/(t^k))$ . 
A: The suggestion in my comment was wrong -- I was incorrectly (and absurdly) extrapolating from Shioda's examples of inseparably uniruled surfaces (which are beautiful, and which I would love to generalize).  Here is the simplest example I know that works in characteristic $0$.
Let $k$ be an algebraically closed field of characteristic $\neq 2,3$, and work in the category of $k$-schemes.  Let $\mathbb{P}_k^4$ have homogeneous coordinates $[Z_0,Z_1,Z_2,Z_3,Z_4]$.  Let $F$ be the degree $4$ (essentially) Fermat polynomial,
$$
F(Z_0,Z_1,Z_2,Z_3,Z_4) = Z_0^4 + Z_1^4 + Z_2^4 + Z_3^4 + Z_4^4.
$$
Denote by $X\subset \mathbb{P}^4_k$ the zero scheme of $F$.  Since $\text{char}(k)\neq 2$, this is a smooth quartic hypersurface of dimension $n=3$.  Now let $U$ be the open subset of $X\times X$ that is the complement of the diagonal $\Delta_X$.  Let $F\subset U$ be the closed subscheme parameterizing ordered pairs of distinct points $([S_0,S_1,S_2,S_3,S_4],[T_0,T_1,T_2,T_3,T_4])$ such that the homogeneous polynomial in coordinates $[S,T]$,
$$
F(TS_0-ST_0,TS_1,-ST_1,TS_3-ST_3,TS_4-ST_4) = 
T^4(S_0^4+S_1^4+S_2^4+S_3^4) - $$
$$
 4ST^3(S_0^3T_0+S_1^3T_1+S_2^3T_2+S_3^3T_3+S_4^3T_4) +
$$
$$
6S^2T^2(S_0^2T_0^2 +S_1^2T_1^2 + S_2^2T_2^2 + S_3^2T_3^2 + S_4^2T_4^2) -
$$
$$
4S^3T(S_0T_0^3+S_1T_1^3+S_2T_2^3+S_3T_3^3+S_4T_4^3) +
$$
$$
S^4(T_0^4+T_1^4+T_2^4+T_3^4+T_4^4).
$$
Since the characteristic is not $2$ or $3$, it is straightforward to see that $F$ has codimension $3$ in $U$.  It follows that the space of lines, $\overline{\mathcal{M}}_{0,0}(X,1)$ in the stable map notation, is a projective curve.  
Now let $i$ be a root of $t^4+1$ in $k$, and consider the hyperplane section $\text{Zero}(Z_4-iZ_3)\cap X$. Using homogeneous coordinates $[Z_0,Z_1,Z_2,Z_3]$ on $\text{Zero}(Z_4-iZ_3)$, the intersection with $X$ is the zero scheme of $Z_0^4+Z_1^4+Z_2^4$.  In particular, there is no dependence on $Z_3$.  This means that the hyperplane section is a cone with vertex $[Z_0,Z_1,Z_2,Z_3] = [0,0,0,1]$.
Thus the plane curve $$
B= \{[Z_0,Z_1,Z_2] \in \mathbb{P}^2_k |Z_0^4+Z_1^4+Z_2^4 = 0\},
$$
is an irreducible component of the space of lines: for every $[S_0,S_1,S_2]$ in $B$, associate the line $L=\text{span}([S_0,S_1,S_2,0,0],[0,0,0,1,i]\}$.  But it is straightforward to compute that the normal bundle of $L$ in $X$ is isomorphic to $\mathcal{O}_L(-2)\oplus \mathcal{O}_L(1)$.  Thus, the Zariski tangent space at every point of $B$ is $2$-dimensional.  Therefore the space of lines is everywhere reduced along $B$.
I believe this example is an exercise in Debarre's textbook.
