On Beauville's conjecture and related topics (Q1905827)

The author proves A. Beauville's conjecture: Let \(f: X\to \mathbb P^1\) be a semistable non-isotrivial fibration with \(g>1\), then the number of singular fibers \(s \geq 5\). The author also proves the strict canonical class inequality: \(K^2_{S/C} < (2g (C)- 2+s)\) \((2g-2)\). But the restriction \(s>0\) cannot be excluded according to the author's proof of theorem 2.5.1. \textit{M.-H. Saito}'s new example for \(s=5\) is reproduced in this paper. In fact these main results are first obtained by \textit{S.-L. Tan} [J. Algebr. Geom. 4, No. 3, 591--596 (1995; Zbl 0864.14003)].