I read the following statement:
Let $X$ be a Del Pezzo surface (i.e. smooth surface with $-K_X$ ample) over an algebraically closed field $k$. Then the general member of $|-K_X|$ is irreducible and reduced.
If $L$ is very ample, the generic smoothness of members in $|L|$ is guaranteed by Bertini's theorem. However, here we only have the ampleness, and I don't know if there a version of Bertini's theorem which states so?