Let $X$ be a projective variety. Let $L$ be an ample line bundle on $X$. I noticed the notation $C\in |L|_s$, and they say that it denotes a smooth curve $C\in |L|$. There are two questions:
a) if $X$ is smooth, then what does $C \in |L|_s$ mean?
b) if $X$ is a variety with finitely many singularities, then what does it mean? Does it mean that it has to avoid the singularities of $X$?
And it seems to me that $|L|_s$ is an open subset of $|L|$. Is that correct?