When is the canonical divisor of an algebraic surface smooth?
Is there some condition on a complex algebraic surface that implies it has a smooth canonical divisor? I am searching for the sharpest possible condition, but sufficient criteria would be nice as well.
For example, the question is quite simple for surfaces that can be embedded in $P^3$. Roughly, just notice that in this case the linear system of the canonical line bundle is base point free and hence, by Bertini's theorem, we can find a smooth canonical divisor.
When we cannot avoid having a singular canonical divisor, then we are left with some singular curves. I am interested in computing the Euler characteristic of their symmetric product. I will post a question about this in a follow up enquire.