The number of singular fibers in hyperelliptic Lefschetz fibrations

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.