3
$\begingroup$

Let $f:X\to \mathbf{P}^1$ be a branched cover of smooth projective connected curves over $\overline{\mathbf{Q}}$, where $\mathbf{P}^1$ is the projective line over $\overline{\mathbf{Q}}$.

What relations are there between the discriminant of the function field extension associated to $f:X\to \mathbf{P}^1$ and other more common "invariants" of the cover like the ramification divisor, etc. ?

  • 0
    $I$ meant (branched) cover.2012-02-27

1 Answers 1

2

The ramification divisor on $X$ is $\sum_x (e_x-1)[x]$ where $e_x$ is the ramification index of $O_{P^1,f(x)}\to O_{X,x}$. The different of the extension $O_{P^1,f(x)}\to O_{X,x}$ is $e_x-1$ (see e.g. Serre, Local fields). The discriminant at $y\in P^1$ is $\sum_{x\in f^{-1}(y)} (e_x-1)$ (Serre, op. cit.).

In summary, if the ramification divisor is $R:=\sum_x (e_x-1)[x]$ on $X$, the discriminant divisor is $\sum_y(\sum_{x\in f^{-1}(y)} (e_x-1))[y]=f_*R$.

It is very important here that the base field has characteristic 0.