Let S be an elliptic surface over a curve C and K a canonical divisor on S. Why is it that K^2 = 0? It should come from the fact that K is the pullback from the canonical divisor of C.
I think that since the fibers have genus 1, the pullback lies inside a fiber, so Kx*Kx = 2g-2 = 0
Does it make any sense?