Let all varieties be projective over an algebraically closed field.
I'm having trouble applying Riemann-Hurwitz formula to a certain morphism $C\to\Bbb{P}^1$, which I define now.
Let $S$ be a rational elliptic surface with elliptic fibration $f:S\to \Bbb{P}^1$. Let $B_1,B_2,B_3$ be bisections of $S$ (i.e. curves on $S$ such that $f|_{B_i}:B_i\to\Bbb{P}^1$ is a flat finite morphism of degree $2$). The three bisections have the following property:
- There is some $t\in\Bbb{P}^1$ such that $f^{-1}(t)$ is tangent to each $B_i$ at $P_i$ with $P_i\neq P_j$ for all distinct $i,j\in\{1,2,3\}$.
- There is some $Q\in S$ such that $B_i\cap B_j=\{Q\}$ for all distinct $i,j\in\{1,2,3\}$.
We define a curve $C:=B_1\times B_2\times B_3$ as the fiber product with the projections $f_i:=f|_{B_i}:B_i\to\Bbb{P}^1$, which gives a projection $p:C\to\Bbb{P}^1$.
I'm trying to compute the genus of the curve $C$, for which I apply Riemann-Hurwitz formula to $p:C\to\Bbb{P}^1$.
I know that $p$ ramifies at $t$, at $s:=f(Q)$ and nowhere else. But the rest is not clear to me. What's the degree of $p:C\to\Bbb{P}^1$? How to compute the ramification indicies over $t$ and $s$?
The first difficulty I have is how characterize the points in $C$. When I think in terms of ordered pairs, then for a general $u\in\Bbb{P^1}$ we have $\#f_i^{-1}(u)=2$ for each $i=1,2,3$, so does that mean that $\#p^{-1}(s)=2\times 2\times 2=8$ hence $\deg p=8$?
In that case, $p^{-1}(t)=\{(P_1,P_2,P_3)\}$? What about $p^{-1}(s)$? How do I compute the ramification indices?
Thank you!