# Tame morphism from a curve to $\mathbb{P}^1$

Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $C$ be a smooth projective curve over $k$. Is it possible to find a map $C \to \mathbb{P}^1$ that is tamely ramified at every point of $C$, i.e. such that the ramification index at every point of $C$ is prime to $p$?

A result of Fulton says that, if $k$ is (algebraically closed) of characteristic $p\ne 2$, then it is possible to find a morphism $C \to \mathbb{P}^1$ that is a simple cover: only double points may appear and at most one in every fiber. (This is theorem 8.1 in "Hurwitz schemes and the irreducibility of moduli of algebraic curves", Ann. of Math. 90, 1969. He says it is classical and dates back to Severi.)

Fulton's result gives a positive answer for fields of characteristic $p\ne 2$. But what about characteristic 2? Does the result still hold? I would already be interested in answers in particular cases (elliptic curves for instance).

EDIT: I added the hypothesis that the field is algebraically closed in order to focus on what I am really interested in. Still, I would also appreciate comments on how relevant this hypothesis is.

• Consider a degree $4$ cover of $\mathbb P^1$ that is ramified over $4$ points, of degree $3$ at each. It is easy to see via the group theory of $S^4$ that such exists. Then by Riemann-Hurwitz, this is an elliptic curve. The moduli space of such covers is one-dimensional, and one just has to check that the map to the moduli space of elliptic curves is nonconstant, which I think is true but haven't checked. Similar explicit constructions might work for other genuses. Apr 15, 2013 at 16:30
• Thanks Will. There are a few things I don't understand. What is the link with $S^4$? How do you know the dimension of the moduli space of covers? (Since $M_{0,4}$ has dimension 1, I would have thought it would be bigger, but this will not cause any trouble anyway.) I don't have time to check the details right now but will come back to it next week. Apr 16, 2013 at 9:17
• @Will: $S^4$ is meant to be the symmetric group on $4$ letters? Anyway, in this characteristic $2$ context without Riemann's existence theorem, it isn't clear to me that this cover exists. I would be more convinced if one would write it down explicitly for an elliptic curve in Weierstrass form. Apr 16, 2013 at 9:29
• @Peter: Thanks for your comment. This $S^4$ makes more sense but I don't know how to make the argument work in characteristic 2 either. On the other hand, I may be able to do it using patching techniques from inverse Galois theory (maybe if $k$ is algebraically closed but I don't mind too much). Apr 16, 2013 at 9:40
• 1) The condition that $k$ be algebraically closed is certainly not necessary in the result from Fulton's paper, it suffices to assume that $k$ be infinite. The result probably holds over finite fields as well, but would require some more arguments. 2) An example of an elliptic curve in characteristic $2$ where there is a tamely ramified map is the elliptic curve with an automorphism of order $3$: the quotient of the curve modulo the group generated by this automorphism is $\mathbb{P}^1$.
– naf
Apr 25, 2013 at 10:51