In Beauville's Complex Algebraic Surfaces, there were some claims about lines on a surface which appeared in proofs, but I can't seem to find justifications for them, and I hope to get some help here.
In the proof of Proposition IV.12, for a del Pezzo surface $S_d$, it's written that
Since $H \equiv -K$, the lines on $S_d$ are just its exceptional curves.
- Is $H$ an arbitrary hyperplane here? How does every line is an exceptional curve follow from $H \equiv -K$?
Also, in the second line of Proposition IV.16, where $S$ is the complete intersection of two quadrics in $\mathbb{P}^4$, the second line of the proof writes:
Any line $E \subset S$ satisfies $E.H=1$, so $E.K = -1$ and $E^2 = -1$.
- How do all these follow from the description of $S$?
This follows from the degree-genus formula: If $ C $ is a smooth curve of geometric genus $ g $ on a surface $ S $ (this can be relaxed to arithmetic genus too, for singular curves, by a deformation argument) then $ 2g -2 = C \cdot (C + K_S) $
In your case if $ C $ is a line $ L $, then $ g=0 $ and $ L.H = 1 $ for a general hyperplane, or rather hyperplane section of $ S $. Since $ K_S = -H $ the formula reads $ -2 = L^2 - 1 $ so $ L^2 = -1 $ showing $ L $ is exceptional.