Let $X$ be a simply-connected smooth projective variety over $\mathbb C$. Let $C$ be a curve on $X$.

If $Y$ is a ramified cover of $X$ of degree $n$, and $D$ is the branch divisor of $Y$, call $(D \cdot C)/n \in \mathbb Q$ the root discriminant of $Y$. Intuitively, this measures the total amount of ramification of the cover $Y$.

Let $r_n$ be the minimal root discriminant among all $Y$ of degree $n$ over $X$.

What is $\lim \inf_{n \to \infty} r_n$?

I want to know how ramified a cover must be. Since low-degree covers can be very exceptional and might have unusually small ramification, I'm asking about covers of sufficiently large degree - thus, a $\lim \inf$.

We can get an upper bound by finding a movable divisor $D$ and a rationally equivalent divisor $D'$ and taking the cyclic cover that adjoins an $n$th root of a function whose zeroes are $D$ and poles are $D'$, whose branch divisor will be $(n-1)(D+D')$. This shows that $r_n \leq 2 (D \cdot C)$ whenever $D$ is movable.

For $X=C=\mathbb P^1$, we can prove that this is optimal, using Riemann-Hurwitz. $r_n=2-2/n$, so $\lim \inf_{n\to\infty} r_n=2$. This suggests the guess:

Are there conditions on $X$ and $C$ that imply that $\lim \inf_{n\to \infty} r_n$ is equal to $2 \min_D (D \cdot C)$, where the minimum is taken over all movable divisors $D$ ?

Unfortunately, I can't verify any higher-dimensional case of this.

This question was motivated by an attempt to find an analogue of a difficult question in number theory. The analogy is that $X$ is like $\mathbb Q$, $Y$ is like a number field, the branch divisor $D$ is like the discriminant of that number field, the intersection number $D \cdot C$ is like the log of the discriminant, so dividing it by $n$ is like the log of the root discriminant. The number theory question is to find the minimal value of the root discriminant among fields of large degree. This is known to be between $22.3$ and $93$, and these bounds turn out to be useful for some interesting applications.

Varieties of dimension $1$ are the most natural analogues of number fields, but this question is solved completely in dimension $1$, where there is just one simply-connected variety. Thus I generalized the question to higher dimensions, using an intersection number $D \cdot C$ to measure the size of the divisor $D$. Other measures of the size of $D$, such as the self-intersection number, might also be interesting.