A note on smooth multiple fibers in pencils of algebraic curves (Q317384)

Let \(f:S \to B\) be a relatively fibred surface, i.e., \(f\) is a surjective morphism from a non-singular projective surface to a non-singular projective curve \(B\) with connected fibers. A fiber \(F\) of \(f\) is called a multiple fiber if there exist a positive integer \(m>1\) and a numerically one-connected effective divisor \(D\) such that \(F=mD\). In this paper, the author studies systematically the multiple fiber of the simplest kind, which is by definition the case when the divisor \(D\) as above is smooth. One of the motivation is to clarify the influence on the gonality and the base locus of the canonical linear system.