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Let $X$ be a smooth proper curve over a number field $K$, and let $f : X\rightarrow\mathbb{P}^1_K$ a degree $n$ map unramified outside $\{0,1,\infty\}$.

I've read in various places that something of the following type is true:

$$\text{$X$ has potential good reduction at all primes $p > n$.}$$

Does anyone have a reference for this?

EDIT: Thanks to Vesselin Dimitrov, this is implied by $\S4,5$ of Beckman's "Ramified Primes in the Field of Moduli of Branched Coverings".

Secondly, is the following known to be true? $$\text{$X$ has potential good reduction at all places away from 2,3, and the primes dividing $n$}$$

Thirdly, could something of the following type be true? $$\text{There exists a finite set $S$ of primes, independent of $X,f$, such that $X$ has potential good}$$ $$\text{reduction at all primes $p\notin S$ which do not divide the ramification indices of $f$}$$ (Ie, does anyone know of a counterexample to such a statement?) In particular I'd be interested in seeing a counterexample in the cases where $S = \emptyset$ and $S = \{2,3\}$.

Note that in both cases I'm asking only about good reduction of $X$, not good reduction of $f$.

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  • $\begingroup$ The more precise condition is that $p$ does not divide the order of the Galois group (which in turn divides $n!$). Did any of the places that gave this statement make a reference to the paper [Ramified primes in the field of moduli of branched coverings of curves, J. Algebra, 1989] by Sybilla Beckmann? It should be possible to read this from Corollary 5.7 at the very end of her paper. $\endgroup$
    – Vesselin Dimitrov
    Commented Sep 10, 2017 at 23:18
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    $\begingroup$ (And you actually need the strict inequality $p > n$ in your first display.) $\endgroup$
    – Vesselin Dimitrov
    Commented Sep 10, 2017 at 23:21
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    $\begingroup$ (Sorry, this is Proposition 5.3, not 5.7, of that paper. Also, by 'Galois group' here I mean the group of the Galois closure of the covering. That means your more precise condition in the first display is indeed right in the case of a Galois covering.) $\endgroup$
    – Vesselin Dimitrov
    Commented Sep 10, 2017 at 23:50

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