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 $f$.

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 $f$.

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:

$$\text{$X$ has potential good reduction at all primes $p > n$. (maybe even $p\nmid n$?)}$$

Does anyone have a reference for this?

Secondly, could something of the following form 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 $S = \{2\}$.

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

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 $f$.

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:

$$\text{$X$ has potential good reduction at all primes $p\ge n$. (maybe even $p\nmid n$?)}$$

Does anyone have a reference for this?

Secondly, could something of the following form 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 $S = \{2\}$.

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

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:

$$\text{$X$ has potential good reduction at all primes $p > n$. (maybe even $p\nmid n$?)}$$

Does anyone have a reference for this?

Secondly, could something of the following form 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 $S = \{2\}$.

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