[Math] Elementary proof of the Hurwitz formula

algebraic-curvesalgebraic-geometryreference-request

I am aware of two forms of the Hurwitz formula. The first is more common, and deals only with the degrees. So if $f:X \rightarrow Y$ is a non-constant map of degree $n$ between two projective non-singular curves, with genera $g_X$ and $g_Y$, then
$$
2(g_X-1) = 2n(g_Y-1) + \deg(R),
$$
where $R$ is the ramification divisor of $f$.
The proof of this was given to me as an exercise when I started my PhD, and I am very happy with it.

However, in some other work that I was doing it appeared that one could strengthen this to say rather that if $K_X={\rm div}(f^*(dx))$ and $K_Y={\rm div}(dx)$ are canonical divisors of $X$ and $Y$, then
$$
K_X = n\cdot K_Y + R.
$$
I have found this alluded to in a number of places, and even stated in Algebraic Curves Over Finite Fields by Carlos Moreno. However, this was without proof, and every idea of a proof that I have seen is in sheaf theoretic language. I am slowly getting through sheaves and schemes, but I am currently trying to prove this in an elementary manner (fiddling around with orders of $dx$ etc.), in the wildly ramified case (the tamely ramified case is fine).

I would like to know if

  1. It is possible to prove this without using sheaves etc in the wildly ramified case
  2. If so, are there any references that would help with this.

edit: Also, is there a different name for the "more specific" Hurwitz formula?

Best Answer

You might like Griffiths's proof of Riemann-Hurwitz (Introduction to Algebraic Curves page 91, theorem (8.5) ) for a morphism $f:X\to Y$ .
It is very natural: he starts from a meromorphic differential form $\omega \in \Omega(Y)$ (whose existence he provisionally admits ) , lifts it to $f^*\omega \in \Omega(X)$, then computes and compares the divisors $div(f^*\omega), f^*(div(\omega))$ and $R$ locally in coordinates.

Riemann-Hurwitz is not difficult but somewhat explains the slightly confusing result that taking divisors and lifting do not commute for differential forms.

Edit
Stichtenoth gives a proof for fields of any characteristic in Corollary 3.4.13 in the language of function fields, but only under a separability assumption.
The most general formula I'm aware of (but I'm not in the least a specialist) and which does not require separability assumptions is Tate's Genus Formula.
You can find it as Corollary 9.5.20 in Villa's Topics in the Theory of Algebraic Function Fields.

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