Talk:Riemann–Hurwitz formula

Examples needed
This page should include examples
 * counting the genus of hyperelliptic curves
 * counting the genus of some superelliptic curves (e.g. y^3 = f(x))
 * relate these to schemes in P^1\times P^1
 * Also mention wild ramification, https://math.berkeley.edu/~mcivor/256B.pdf — Preceding unsigned comment added by Wundzer (talk • contribs) 18:22, 29 April 2020 (UTC)

A possible mistake in the formula $$\chi(S')- r = N \cdot (\chi(S) - b) $$
It seems to me that the formula $$\chi(S')- r = N \cdot (\chi(S) - b) $$ and its proof are wrong as stated, since the fiber of a branch point may contain unramified points, and thus the restriction of $$\pi$$ to the unramified points as stated in the proof is undefined. Any univariate polynomial whose derivative has more than one distinct root provides a counterexample when considered as a map of Riemann spheres.

To correct the error, it seems sufficient to replace $$r$$ by the size of the union of the fibers of the branch points, which contains all the ramification points together with perhaps some unramified points, denote this quantity by $$b'$$. In this formulation, the proof works as stated in the article, and we can also deduce it from the usual formulation of Riemann-Hurwitz, as we have $$N \cdot b - b' = \sum_{P\in S'} (e_P - 1)$$ since for any $$Q \in S$$ we have $$N = \sum_{P \in \pi^{-1}(Q)} e_P$$ Horrific Necktie (talk) 09:36, 12 August 2022 (UTC)