Fibers of pencils of curves on smooth surfaces

Let $X$ be a smooth projective surface such that linear and numerical equivalence of divisors on $X$ coincide and let $\sigma \subseteq \mathrm{\mid }D\mathrm{\mid }$ be a linear pencil on $X$ with integral general fibers. A fiber of $\sigma$ will be called special if either it is not integral or it has non-generic multiplicity at some of the base points (including the infinitely near ones) of the pencil. In this note we provide an algorithm to compute the integral components of the special fibers of $\sigma$.