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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.

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Did you mean for $Y$ to be in some way related to $C$ as a cover of $X$? –  KConrad Feb 16 at 21:37
    
@KConrad: No. $C$ is just there to turn $D$ into a number. –  Will Sawin Feb 16 at 22:00

1 Answer 1

When $X$ is a curve, the Riemann-hurwitz formula is the only condition on $D$. Once $X$ has positive genus, $\deg D$ can be any even number, including zero. For example, if $X$ is an elliptic curve then there are isogenies $X' \rightarrow X$ of arbitrary degree; more generally $X$ embeds into its Jacobian, and the fiber product of the embedding $X \hookrightarrow J(X)$ with any isogeny $\phi: A \rightarrow J(X)$ gives an unramified cover of $X$ whose degree equals $\deg\phi$. When $X$ has genus at least $2$ one can also construct non-abelian unramified covers from suitable maps from $\pi_1(X)$ to finite non-abelian groups.

In particular, if $X$ has positive genus then $\liminf_n r_n = 0$.

(Perhaps the OP knows this already, but it doesn't seem to be stated anywhere in the question statement or the comments thus far.)

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I indeed knew this, which is why I assumed in the question that $X$ is simply connected. But it's probably good to state this explicitly. Indeed, this reminds me of a useful technique for upper bounding $\lim \inf r_n$: If $Y$ is cover of $X$ with root discriminant $r$ and has infinite etale fundamental group, then all etale covers of $Y$ are ramified covers of $X$ with root discriminant $r$, so $\lim \inf_n r_n \leq r$. –  Will Sawin Feb 17 at 1:44
    
In fact, one can use this to show that $\lim \inf_n r_n$ is not necessarily nonnegative: Take $A$ an abelian surface, $X$ a Kummer surface created by blowing up $A/\{\pm 1\}$, $C$ one of the exceptional $-2$ curves from this blowup. Let $Y$ be the blowup of all the $2$-torsion points of $A$, then $Y$ is a double cover of $X$ ramified only at the $16$ exceptional curves, so $D \cdot C= C\cdot C=-2$, so $r=-1$, and $\lim\inf_n r_n \leq -1$! –  Will Sawin Feb 17 at 1:47
    
Right, "simply connected" excludes curves of positive genus. –  Noam D. Elkies Feb 17 at 1:54
    
You can leverage Noam's examples into examples with $X$ simply connected yet $\liminf_n r_n$ equal to $0$. For instance, let $X$ be the blowing up of $\mathbb{P}^2$ at the $d^2$ basepoints of a general pencil of degree $d$ plane curves, and let $C$ be a fiber of the projection $\pi:X\to \mathbb{P}^1$. For the generic fiber $X_\eta$, for every integer $m$, associated to the multiplication map $[m]:J(X_\eta)\to J(X_\eta)$, there is an associated branched cover $X_m \to X$ that is branched only over the singular fibers of $\pi$. –  Jason Starr Feb 17 at 14:22
    
@JasonStarr: Thanks for the idea! Note that there is another branched cover that is ramified over the singular fibers - take some branched cover of $\mathbb P^1$ ramified over the singular points, and pull it back. So this seems very similar to the upper bound of $2(D \cdot C)$, for $D$ movable, in this case a fiber. In the $d=3$/elliptic surface case I computed that the pullback construction gives you better upper bounds. Do you think the multiplication-by-$m$ construction might do better for higher genus curves? –  Will Sawin Feb 18 at 2:27

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