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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 $H^i(C, \mathcal{O}_C)$, T_C)\$, i.e. they are the sheaves

`$R^i p_*\mathcal{O}_U$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.

1

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 $H^i(C, \mathcal{O}_C)$, i.e. they are the sheaves

`$R^i p_*\mathcal{O}_U$`

where `$p : U \to \overline{\mathcal{M}_{g,n}}$` is the universal family. 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.
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