TITLE: a formula about conic bundles
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: 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 by de Jong and myself.