Deforming curves to other curves over the field of rational numbers

Question

Let $X$ and $Y$ be smooth projective geometrically connected curves over $k$ of genus $g$ at least two.

If $k$ is an algebraically closed field of characteristic zero, there exists a connected variety $T$ over $k$, points $x,y \in T(k)$ and a family of curves $\mathcal C\to T$ such that $\mathcal C_{t_0} = X$ and $\mathcal C_{t_1} = Y$. In other words, there is a deformation space over $k$ which deforms $X$ into $Y$.

My question is whether such a space exists if $k$ it not algebraically closed.

Suppose that $k = \mathbb Q$. Does there exist a deformation from $X$ to $Y$ over $k$?

More precisely, does there exist a connected variety $T$ over $\mathbb Q$, points $x,y\in T(k)$ and a family of curves $\mathcal C\to T$ such that $\mathcal C_{t_0} = X$ and $\mathcal C_{t_1} = Y$?

When $X$ and $Y$ have level $n$ structure (for some $n>2$), then you can take $T$ to be the moduli space of curves with level n structure over $\mathbb Q$. Also, without any assumption of level structure, it is clear that there exists such a deformation space over a finite extension of $\mathbb Q$. But does it exist over $\mathbb Q$?

My guess would be that such a space doesn't exist in general.

Answer 1

Yes. Let $d\geq 3$ be an integer. Define $N$ to be $(2d-1)(g-1)$. Denote by $P(t)$ the Hilbert polynomial $2d(g-1)t + 1-g$. Let $H^{P(t)}_{\mathbb{P}^N_k/k}$ denote the Hilbert scheme parameterizing closed subschemes $C$ of $\mathbb{P}^N_k$ with Hilbert polynomial $P(t)$. By Grothendieck, this exists and is a projective scheme over $k$.
Let $T$ denote the locally closed subscheme of $H^{P(t)}_{\mathbb{P}^N_k/k}$ parameterizing closed subschemes $C$ that are smooth, integral curves with $\mathcal{O}_{\mathbb{P}^N}(1)|_C$ isomorphic to $\omega_{C/k}^{\otimes d}$. Using infinitesimal deformation theory, $T$ is smooth over $k$ of the "expected" relative dimension. Using irreducibility of the moduli space, $T$ is also geometrically integral. Denote by $\mathcal{C} \to T$ the restriction over $T$ of the universal closed subscheme of $\mathbb{P}^N_k$. By choosing bases of $H^0(X,\omega_{X/k}^{\otimes d})$ and $H^0(Y,\omega_{Y/k}^{\otimes d})$, you can find $k$-points, $x$ and $y$, of $T$ whose fibers, $\mathcal{C}_x$ and $\mathcal{C}_y$, are isomorphic to $X$ and $Y$.