Let $\pi:Z\to S$ be a conic bundle over a smooth complex surface $S$. I'd like to know how to prove that $-\pi\_{\*}K\_{Z}^{2}=4K\_{S}+\Delta$, where $\Delta$ denotes the locus in $S$ over which the fibres are singular.



a formula about conic bundles
There are a number of sources, or you could just prove this for yourself. In the context of algebraic geometry, this follows from Proposition 5.1(v) of [Divisor Classes and The Virtual Canonical Bundle for Genus 0 Curves](http://arxiv.org/pdf/math/0602642v1.pdf) by de Jong and myself.


