The number of singular fibers in hyperelliptic Lefschetz fibrations

Abstract

We consider complex surfaces, viewed as smooth 4-dimensional manifolds, that admit hyperelliptic Lefschetz fibrations over the 2-sphere. In this paper, we show that the minimal number of singular fibers of such fibrations is equal to 2𝑔 + 4 for even 𝑔 ≥ 4. For odd 𝑔 ≥ 7, we show that the number is greater than or equal to 2𝑔 + 6. Moreover, we discuss the minimal number of singular fibers in all hyperelliptic Lefschetz fibrations over the 2-sphere as well.