Suppose that $S$ is a smooth complex projective surface ($\mathbb C$-scheme, reduced, irreducible...).
What do algebraic geometers usually mean with the term a smooth family of divisors in $S$?
In general with the notation family $\mathcal F$ over an "object" $X$, I intend that there exists a morphism $f:Y\longrightarrow X$ (where $Y$ "is an object of the same type of $X$") such that every fiber $Y_x:=f^{-1}(x)$ is an element of the family $\mathcal F$.
So in my case do I have to find a variety $Y$ and a morphism $f:Y\longrightarrow S$ such that every fiber $Y_s$ is a divisor of $S$? Moreover such a morphism $f$ is supposed to be smooth...