On the slope of hyperelliptic fibrations with positive relative irregularity

Abstract

Let $f:S\to B$ be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus $g\ge 2$ with relative irregularity $q_f$. We show a sharp lower bound on the slope ${\lambda }_f$ of $f$. As a consequence, we prove a conjecture of Barja and Stoppino on the lower bound of ${\lambda }_f$ as an increasing function of $q_f$ in this case, and we also prove a conjecture of Xiao on the ampleness of the direct image of the relative canonical sheaf if ${\lambda }_f<4$.

final version, accepted by Trans. Amer. Math. Soc

References in corpus (2)

• The Oort conjecture on Shimura curves in the Torelli locus of curves
• On Shimura curves in the Torelli locus of curves

Cited by in corpus (2)

• On the slope conjecture of Barja and Stoppino for fibred surfaces
• Xiao's conjecture for general fibred surfaces