Let $C$ be a smooth projective curve over $\mathbb{C}$ with automorphism group $G$. When is it possible to construct an analytic family of smooth projective curves $\mathcal{C} \longrightarrow T$ (e.g. $T = \mathbb{C}$) such that $\text{Aut }C\_t \cong G$ for (almost) every $t \in T$ ($C\_t$ is the fiber over $t \in T$)? 


I came across a paper that claimed that this is always possible when $G$ is a cyclic group of order $p$ for some odd prime $p$, but I'm not sure what kind of construction the author had in mind or whether this can be generalized. Also, what happens if we replace the condition $\text{Aut }C\_t \cong G$ by $|\text{Aut }C\_t| \ge G$?



Family of smooth projective curves with given automorphism group
You might also want to have a look at a book about Hurwitz stacks by Bertin-Romagny called "[Champs de Hurwitz](https://perso.univ-rennes1.fr/matthieu.romagny/articles/champs_de_Hurwitz.pdf)". There they construct spaces $\mathcal{H}\_{g,G,\xi}$ of smooth projective curves $C$ with an action of a finite group $G$ and specified fixed-point behaviour recorded in $\xi$. Since (I guess) automorphism groups in a family of curves jump in size on closed subsets of the base, you can construct a family as you desire by taking the open subset of $\mathcal{H}\_{g,G,\xi}$ where the automorphism group of $C$ is exactly $G$. In a certain sense this is the universal such family.


The dimension of the space $\mathcal{H}\_{g,G,\xi}$ is exactly $3g'-3+b$ where $g'$ is the genus of the quotient curve $C/G$ and $b$ is the number of branch points. So just as Jason said, the space is positive-dimensional unless $g'=0$ and $n=3$. 


