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This question concerns a statement from the paper by Ben Moonen "Special subvarieties arising from families of cyclic covers of the projective line. Documenta Math. 15 (2010)", Lemma 5.5. ii).

More precisely, Let $C\to T$ be a family of cyclic covers of $\mathbb{P}^{1}_{\mathbb{C}}$. Here $T$ can be takec to be the big diagonal in $(\mathbb{A}^{1})^N$, i.e., $T=\{(t_1,\cdots, t_N)|t_i\neq t_j \text{ for } i\neq j \}$.
Take a $t\in T(k)$, We want to show the following:

Consider the equivariant Kodaira-Spencer map $\rho^G: T_t\to H^1(C,\Theta_{C/T})^G$ (where $G=\mu_m$ is the cyclic group of the covering). Show that $\rho^G$ is surjective, in other words show that the equivariant deformation is complete in the sense that if $C_t$ is a fiber of $C\to T$, then we must show that a G-equivariant deformation $D$ of $C_t$ comes actually from the family $C\to T$ by pull-back. The author then says that this is obvious because $D/\mu_m\cong \mathbb{P}^{1}_{k[\epsilon]}$. I don't quite understand why this isomorphism alone proves the surjectivity.

My argument is as follows:

There is an isomorphism $Def(C_t,G)\cong Def(\mathbb{P}^{1},B)$ between G-equivariant deformations of $C_t$ and deformation of $\mathbb{P}^{1}$ with marked points in $B=$set of branch points of $C_t\to \mathbb{P}^{1}$. Now since $D/\mu_m\cong \mathbb{P}^{1}_{k[\epsilon]}$ and since $\mathbb{P}^{1}_{k[\epsilon]}$ is a deformation of $\mathbb{P}^{1}$ (considered with marked points in $B$) by the above isomorphism, it corresponds to a G-equivariant deformation of $G$.

Is this argument correct? If not can someone explain the the surjectivity of equivariant Kodaira-Spencer map?

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