Abstract
We prove that for a hyperelliptic fibration on a surface of general type with irreducible fibers over a (possibly) non-complete curve, the image of the fundamental group of a general fiber in the fundamental group of the surface is finite. Examples show that the result is optimal. As a corollary of this result we prove two conjectures; the Shafarevich conjecture on holomorphic convexity for the universal cover of these surfaces, and a conjecture of Nori on the finiteness of the fundamental groups of some surfaces. We also prove a striking general result about the multiplicities of multiple fibers of a hyperelliptic fibration on a smooth, projective surface.
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Gurjar, R.V., Paul, S. & Purnaprajna, B.P. On the fundamental group of hyperelliptic fibrations and some applications. Invent. math. 186, 237–254 (2011). https://doi.org/10.1007/s00222-011-0318-7
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DOI: https://doi.org/10.1007/s00222-011-0318-7