On the slope of hyperelliptic fibrations with positive relative irregularity

الأصل · EN

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