TITLE: on a Deformation long exact sequence of moduli space of stable maps
QUESTION [7 upvotes]: I am reading the book "mirror symmetry" by Hori,Katz,Klemm,etc. And I want to understand the following Deformation long exact sequence
\begin{align}
 0 & \to Aut(Σ, p_1, . . . , p_n, f)\to Aut(Σ, p_1, . . . , p_n) &\newline \to Def(f) &→ Def(Σ, p_1, . . . , p_n, f) → Def(Σ, p_1, . . . , p_n)  &\newline \to 
Ob(f) &\to Ob(Σ, p_1, . . . , p_n, f) \to 0
\end{align}
it connects three deformation theory:
1. deformation of stable curves
2. deformation of maps(with fixed source)
3. deformation of stable maps(with possible changing source curves)
And my understanding goes as follows:
Let $\mathscr{X}=M_{g,n}$ be the moduli stack of algebraic curves(genus $g$, n-marked point), and let $\mathscr{Y}=M_{g,n}(X,\beta)$ be the moduli stack of stable maps. Then there is a natural "forgetful" morphism:
$\pi : \mathscr{Y} \to \mathscr{X}$
by forgeting the "map".  
We have a distinguished triangle of cotangent complexes in the derived category $D^{-} (\mathscr O_{\mathcal{Y}})$: 
\begin{equation}
 \pi^* L_{\mathscr{X}}\to L_{\mathscr Y}\to  L_{\mathscr{Y}/\mathscr{X}}\to \cdot
\end{equation}
Now apply $R\mathscr{Hom}$, we have a long exact sequence:
\begin{align}
\mathscr Ext ^0 (L_{\mathscr Y/\mathscr X },\mathcal O_{\mathscr Y }) & 
\to \mathscr Ext^0 (L_{\mathscr Y}, \mathcal O_{\mathscr Y} ) \to \mathscr Ext^0 (\pi^* L_{\mathscr X},\mathcal O_{\mathscr Y} )& \newline
\to \mathscr Ext ^1 (L_{\mathscr Y/\mathscr X },\mathcal O_{\mathscr Y }) & \to \mathscr Ext^1 (L_{\mathscr Y}, \mathcal O_{\mathscr Y} ) \to \mathscr Ext^1 (\pi^* L_{\mathscr X},\mathcal O_{\mathscr Y} )& \newline
\to \mathscr Ext ^2 (L_{\mathscr Y/\mathscr X },\mathcal O_{\mathscr Y }) & \to \mathscr Ext^2 (L_{\mathscr Y}, \mathcal O_{\mathscr Y} ) \to \mathscr Ext^2 (\pi^* L_{\mathscr X},\mathcal O_{\mathscr Y} )
\end{align}
My questions:
 (1). is it an exact sequence of sheaves on $\mathscr Y$ with the first long exact sequence as its stalks?
 (2). If (1) is true, then how to see $\mathscr Ext^i (\pi^* L_{\mathscr X},\mathcal O_{\mathscr Y} )$ (i=0,1,2) corresponds to Aut,Def,Ob of curves? And why the two ends of the exact seqence vanishes?

REPLY [2 votes]: I don't believe that this is correct. The easiest way to see this is to look at your second question: The automorphisms/deformations/obstructions of a curve come from $H^i(C, T_C)$, i.e. they are the sheaves
$R^i p_*\omega_{U/\overline{\mathcal{M}_{g,n}}}^\vee$
where $p : U \to \overline{\mathcal{M}_{g,n}}$ is the universal family, and $\omega_{U/\overline{\mathcal{M}_{g,n}}}$ the relative dualizing sheaf. But these do not depend on $\overline{\mathcal{M}_{g,n}}(X, \beta)$ !
In the end, I think the issue is that you have the wrong exact sequence. What you want (to produce the relative obstruction theory) is the complex
$R^i p_*f^*T_X$
where the maps $p, f$ arise in the universal diagram 
$\overline{\mathcal{M}_{g,n}}(X, \beta) \longleftarrow_p U \longrightarrow_f X$
It is not obvious to me that your sheaves should be the same as these ones.