On the minimal number of singular fibers in Lefschetz fibrations over the torus

Abstract

Abstract. We show that the minimal number of singular fibers N (g, 1) in a genus-g Lefschetz fibration over the torus is at least 3. As an application, we show that N (g, 1) ∈ {3, 4} for g ≥ 5, N (g, 1) ∈ {3, 4, 5} for g = 3, 4 and N (2, 1) = 7.

The existence of an indecomposable minimal genus two Lefschetz fibration

“…Remark 13. For g ≥ 3 and h = 1, 2 we find that genus-g Lefschetz fibrations over Σ h constructed in [19,17,29] are indecomposable and minimal. The minimality follows from the result of [27], and the indecomposability follows from the number of singular fibers and the lower bounds on the number of singular fibers of Lefschetz fibrations over S 2 (see [8]) and T 2 (see [29]).…”

“…For g ≥ 3 and h = 1, 2 we find that genus-g Lefschetz fibrations over Σ h constructed in [19,17,29] are indecomposable and minimal. The minimality follows from the result of [27], and the indecomposability follows from the number of singular fibers and the lower bounds on the number of singular fibers of Lefschetz fibrations over S 2 (see [8]) and T 2 (see [29]). In [9], it was shown that a genus-g Lefschetz fibration over Σ h with a "maximal section" (see [9] for the definition) is indecomposable (as a fiber sum of two genus-g Lefschetz fibrations with a section) if h ≥ 1.…”

“…Note that a maximal section means that a (−1)-section for h = 0. If g ≥ 5, then we can show that the fibrations given in [19,17,29] has a maximal section from the constructions of [19,29] and Theorem 13 and the technique in Section 3.3 of [9]. On the other hand, our indecomposable minimal genus-2 Lefschetz fibration over S 2 does not admit any maximal sections (i.e.…”