Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $\sigma_i$ ,$i=1 \dots n$. None of these "sections" are honestly a section, they are actually birational to the base manifold. More precisely, they wrap around a finite number of (-1)-curves, so they correspond to a blow of the base at some smooth points.
Let's blow up the base at those finite number of points, such that at least one of the sections, say $\sigma_1$, becomes isomorphic to the base. So basically, we define a new elliptically fibered threefold $\tilde{X} = X \times_B \sigma_1$.
What I want to do is to compute the intersection of divisors after the blow up in terms of the similar objects in X.
For example, suppose $\sigma_1$ wrap around only one (-1)-curve, I know $\sigma_1$ satisfy relations like,
$\sigma_1^2=-c_1(B)\cdot \sigma_1 + e$,
where $c_1(B)$ is the first Chern class of the base, and $e$ is a codimension 2 cycle, which actually correspond to the (-1)-curve that $\sigma_1$ wrap around. (we can find a relation between e and $\sigma_i D_b$ and $D_b^2$ ($D_b$ is a base divisor))
So, if $\tilde{\sigma_1}$ is the corresponding section in $\tilde{X}$, what can be said about,
$\tilde{\sigma_1} \cdot \tilde{\sigma_1}=?$.
I did some calculations which seem pretty reasonable to me, but the final answers are not satisfying. For example, I compute the Euler number of $\tilde{\sigma_1}$, and it's not correct.
I cannot get into more details, because it will become too lengthy. I would appreciate if someone can give me hint...
