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.


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.

  • $\begingroup$ Thank you very much. I have also found other sources such as Mori and Mukai's paper "On Fano 3-folds with B_{2}>=2",Adv. Stud. Pure Math. 1 101--129(1983). $\endgroup$
    – Jiang
    Nov 5 '14 at 11:17

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