Slopes of Fibered Surfaces with a Finite Cyclic Automorphism

Upper bounds on the slope of certain fibered surfaces

“…Here, a fibration is called bielliptic if a general fiber is a bielliptic curve, i.e., a non-singular projective curve obtained as a double covering of an elliptic curve. In [7], we established the lower bound of the slope for such fibrations of type (g, h, n) extending former results for n = 2 in [2] and [6]. Furthermore, when h = 0 and n ≥ 4, we obtained even the upper bound (expressed as a function in g and n) which is strictly smaller than 12.…”

“…In this section, we recall and state basic results for primitive cyclic covering fibrations in [7]. Definition 1.1.…”

“…The purpose of the present article is to give an affirmative answer to the above mentioned upper bound problem by introducing numerical invariants attached to the singularities of the branch locus of the cyclic covering, and improving a coarser estimates in [7]. When h = 0, a vertical component of the branch locus on a relatively minimal model is always a non-singular rational curve and this fact makes it much easier to handle singularities on the branch locus.…”

“…In [5], upper bounds for genus 3 fibrations are studied from another point of view. We studied in [7] primitive cyclic covering fibrations of type (g, h, n). Roughly speaking, it is a fibered surface of genus g obtained as the relatively minimal model of an n sheeted cyclic branched covering of another fibered surface of genus h. Note that hyperelliptic fibrations are nothing more than such fibrations of type (g, 0, 2) and that bielliptic fibrations of genus g ≥ 6 are those of type (g, 1, 2) (cf.…”

“…The organization of the paper is as follows. In §1, we recall basic results from [7] and [8] on primitive cyclic covering fibrations and introduce some notation for the later use. In §2, we observe the local concentration of relative invariants of primitive cyclic covering fibrations of type (g, 1, n) on a finite number of fiber germs and show Theorem 0.1.…”