On the Slope of Hyperelliptic Lefschetz Fibrations and the Number of Separating Vanishing Cycles

Details On the Slope of Hyperelliptic Lefschetz Fibrations and the Number of Separating Vanishing Cycles On the Slope of Hyperelliptic Lefschetz Fibrations and the Number of Separating Vanishing Cycles

2008-10-07

arxiv.org/abs/0810.1088v2

Mathematics

Symplectic Geometry

23 Pages, Corollary 19 and Corollary 23 added

Scientific paper

In this article we find an upper and lower bound for the slope of genus g hyperelliptic Lefschetz fibrations, which is sharp when g = 2, and demonstrate the strong connection, in general, between the slope of hyperelliptic genus g Lefschetz fibrations and the number of separating vanishing cycles. Specifically, we show that the slope is greater than 4-4/g if and only if the fibration contains separating vanishing cycles. We also improve the existing bound on s/n, the ratio of number of separating vanishing cycles to the number of non-separating vanishing cycles, for hyperelliptic Lefschetz fibrations of genus g>1. In particular we show that s<=n for such fibrations when g>5.

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