TITLE: Intersection with canonical divisor
QUESTION [5 upvotes]: Let $X\longrightarrow \mathbb P^1$ be a rational ellipic surface.
In the Paper "pencils of cubic curves and rational elliptic surfaces" (by C.T.C.Wall) it is stated that C.K < 0, where C is a curve not contained in a fiber and K ist a canonical divisor on X. I cannot see this, can anyone help me?

REPLY [2 votes]: The canonical class of a rational elliptic surface is the negative of the fibre class.  Now consider any irreducible curve $C \subset X$, and its image under $X \to \mathbb{P}^1$.  This image is either a point, in which case $C$ is contained in a fibre, or all of $\mathbb{P}^1$, in which case $C$ must intersect the fibre class positively.