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 $|\text{Aut }C_t| \ge G$?

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    $\begingroup$ The local deformation space of smooth, projective curves over $\mathbb{C}$ together with a specified group $G$ of automorphisms is smooth with Zariski tangent space equal to $H^1(X,T_X)^{G}$, the $\mathbb{C}$-vector subspace of $G$-invariant elements. It is an exercise in Arbarello-Cornalba-Griffiths-Harris that this vector space is zero-dimensional if and only if the quotient $C/G$ is $\mathbb{P}^1$ and the quotient morphism, $C\to C/G,$ is branched over precisely $3$ points (i.e., a Belyi cover). $\endgroup$ – Jason Starr Oct 23 '17 at 17:02
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    $\begingroup$ It looks to me like there are plenty of Belyi covers that are Galois with cyclic Galois group of order $p$, e.g., for homogeneous coordinates $[x,y]$ on $\mathbb{P}^1$, desingularizations of the plane curves $\text{Zero}(z^p-x^ay^b(x-y)^c) \subset \mathbb{P}^2_{\mathbb{C}}$ where $a+b+c$ equals $p.$ $\endgroup$ – Jason Starr Oct 23 '17 at 17:14

You might also want to have a look at a book about Hurwitz stacks by Bertin-Romagny called "Champs de Hurwitz". 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 $n=3$.

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    $\begingroup$ Something that I forgot to say is that the morphism from $\mathcal{H}_{g,G}$ to $\mathcal{M}_g$ is finite (proper and quasi-finite) for $g\geq 2$. Properness can be deduced from the Weil extension result (or by analyzing the graphs of automorphisms in the self-product of the curve). $\endgroup$ – Jason Starr Oct 26 '17 at 19:20

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