Covers with fixed ramification

Question

I have a smooth, projective variety $Y$ and fix a divisor $D \subset Y$ (not assumed irreducible). Is it possible that there exists a non-constant family $f_t : X_t \to Y$ such that each $X_t \to Y$ is a finite map ramified only over $D$? (Feel free to assume char 0, and interpret "family" however is convenient.)

Answer 1

This is not an Answer, but too long for a comment.

We can do the following if $D$ is the support of an ample line bundle $\mathcal L=\mathcal O(D)$. Since $\mathcal L$ is ample, there is some $n\in\mathbb N$ such that $\mathcal L^{\otimes n}=\mathcal O(D)^{\otimes n}=\mathcal O(n\cdot D)$ is very ample - since $n\cdot D$ has the same support as $D$, we may replace $\mathcal L$ by $\mathcal L^{\otimes n}$ and assume that it is very ample.

Let $\phi:Y\to\mathbb P^r$ be the closed immersion induced by $\mathcal L$. Assume that we have a family of branched coverings $g_t:\mathbb P^r\to\mathbb P^r$ where every $g_t$ branches only along the coordinate hyperplanes $\operatorname{Z}_+(X_i)\subseteq \mathbb P^r$ and consider the fiber product

$$\begin{matrix} X_t&\overset{\psi_{t}}\longrightarrow & \mathbb P^r \\ {\scriptstyle f_t}\downarrow~~ & {\scriptstyle\times} & ~\downarrow{\scriptstyle g_t}\\ Y & \underset{\phi}\longrightarrow & \mathbb P^r \end{matrix}$$

Then, $f_t$ does what you want. Indeed, since $\phi$ is injective, so is $\psi_t$. Since $D$ is precisely the preimage of the union of all coordinate hyperplanes in $\mathbb P^r$, this implies that $f_t$ branches precisely along $D$.

Such a $g_t$ can be obtained by taking any family of unramified morphisms $h_t:\mathbb P^r\to\mathbb P^r$, possibly just a family of linear transformations $h_t\in\mathbb P\operatorname{GL}_r$ that do not leave $\phi(Y)$ fixed. I am reasonably sure such a family always exists. Let $h_t=[h_t^1:\ldots:h_t^r]$ be given by certain homogeneous polynomials $h_t^i\in\Bbbk[X_0,\ldots,X_r]$. Then, you can choose ramification indices $d_i\in\mathbb N$ as you please and set $g_t^i := X_i^{d_i}\cdot h_t^i$. The corresponding morphism $g_t:\mathbb P^r\to \mathbb P^r$ branches along the $i$-th coordinate hyperplane.

What if $D$ is not the support of something ample? Well, in this case I am not so sure. You might be able to do something similar, but I don't know off the top of my hat how to do it.