2 fixed type

Let $\overline{\mathcal{M}}_{g,n}(X,\beta)$ be the moduli stack of stable maps $f$ from genus $g$, $n$-marked curve $C$ to a variety $X$ i nth curve class $\beta \in H^2(X,\mathbb{Z})$. I would like to know why the hypertext hyperext group $$Ext^1_C(f^*\Omega_X\rightarrow \Omega_C(\sum_{i=1}^n),\mathcal{O}_C)$$ represents tangent space of the moduli and the obstruction lies in
$$Ext^2_C(f^*\Omega_X\rightarrow \Omega_C(\sum_{i=1}^n),\mathcal{O}_C)?$$ I am aware that the deformation-obstrcution of a map $f:C\rightarrow X$ with a fixed curve $C$ is governed by $H^i(C,f^*T_X)$ for $i=1,2$ and the automorphism-deformation of a $n$-marked points $(C;p_1,\dots,p_n)$ is governed by $Ext^i_C(\Omega_C(\sum_{i=1}^n),\mathcal{O}_C)$ for $i=0,1$. However, I don't know how to combine them into one package in the hyperext groups above.

I would appreciate it if someone could kindly explain how to obtain and understand the hyperext groups above.

1

# Tangent space of the stack $\overline{\mathcal{M}}_{g,n}(X,\beta)$.

Let $\overline{\mathcal{M}}_{g,n}(X,\beta)$ be the moduli stack of stable maps $f$ from genus $g$, $n$-marked curve $C$ to a variety $X$ i nth curve class $\beta \in H^2(X,\mathbb{Z})$. I would like to know why the hypertext group $$Ext^1_C(f^*\Omega_X\rightarrow \Omega_C(\sum_{i=1}^n),\mathcal{O}_C)$$ represents tangent space of the moduli and the obstruction lies in
$$Ext^2_C(f^*\Omega_X\rightarrow \Omega_C(\sum_{i=1}^n),\mathcal{O}_C)?$$ I am aware that the deformation-obstrcution of a map $f:C\rightarrow X$ with a fixed curve $C$ is governed by $H^i(C,f^*T_X)$ for $i=1,2$ and the automorphism-deformation of a $n$-marked points $(C;p_1,\dots,p_n)$ is governed by $Ext^i_C(\Omega_C(\sum_{i=1}^n),\mathcal{O}_C)$ for $i=0,1$. However, I don't know how to combine them into one package in the hyperext groups above.

I would appreciate it if someone could kindly explain how to obtain and understand the hyperext groups above.