Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree

2013-01-07T10:00:29Z

Let $X$ be a regular noetherian scheme of dimension one. Let $d$ be an integer.

Question. Are there only finitely many finite etale morphisms $Y\to X$ of degree $d$?

I want to exclude finite etale morphisms obtained from the base field (if there is any), i.e., if $X$ is of finite type over a field $k$, I don't want to consider the finite etale covers $X_K\to X$ obtained from finite separable field extensions $k\subset K$.

Even if we exclude these finite etale covers, the answer is negative when $X=\mathbf{A}^1_{\mathbf F_p}$.

Then again, the answer is positive if $X$ is

an open subscheme of Spec $O_K$, with $O_K$ the ring of integers of a number field;
a (not necesarily compact) algebraic curve over an algebraically closed field of characteristic zero.
An algebraic curve over a field $k$ of characteristic zero (if we exclude the finite etale covers coming from finite extensions $k\subset K$).

In general the answer is negative as shown above. Which condition on $X$ can we impose to obtain a positive answer.

I was thinking about $X$ has to be of characteristic zero. Does this suffice?

Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree - MathOverflow

Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree