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> Is there a general formula for the canonical divisor $K\_X$ of a smooth conic bundle $X$?
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Motivation: for smooth hypersurfaces of degree $d$ in $\mathbb{P}^n$, $K\_X = \mathcal{O}\_X(d-n-1)$. But smooth conic bundles are not "naturally" embedded in some $\mathbb{P}^n$, so I was wondering if an analogue of the above formula holds for them.



Canonical divisor for conic bundles
Here is a method which should give you what you want, but I leave the details to you as I don't have time to work them out now.


Without loss of generality we may work over an algebraically closed field. Each singular fibre consists of a union of two $(-1)$-curves meeting at a single point. Contracting a choice of $(-1)$-curve in each single fibre, we obtain a ruled surface over $\mathbb{P}^1$. There is a formula for the canonical divisor of a ruled surface, which when combined with the formula for the canonical divisor of a blow-up, should give you what you want (all these tools can be found in Ch. V of Hartshorne).


Note that a useful check is given by Noether's formula, which implies that
$$(K\_X)^2 = 8 - \mbox{number singular of fibres}.$$


