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The minimal number of singular fibers of a semistable curve over $$\mathbb{P}^ 1$$. (English) Zbl 0864.14003
The purpose of this paper is to try to answer L. Szpiro’s following question [Astérisque 86, 44-78 (1981; Zbl 0517.14006)]. Let $$f:S\to \mathbb{P}^1_\mathbb{C}$$ be a family of semistable curves of genus $$g$$, which is not trivial. Then, what is the minimal number of the singular fibers of $$f$$?
A. Beauville gives a lower bound for the number of singular fibers [see Astérisque 86, 97-108 (1981; Zbl 0502.14009)]: With the notations as above, if $$g\geq 1$$, then $$f$$ admits at least 4 singular fibers. – In fact, Beauville conjectured that for $$g\geq 2$$, there is no such fibration with 4 singular fibers. The main result of this paper is:
Theorem 1 (Beauville’s conjecture). If $$f:S\to \mathbb{P}^1_{\mathbb{C}}$$ is a nontrivial semistable fibration of genus $$g\geq 2$$, then $$f$$ admits at least 5 singular fibers.
Theorem 1 is an immediate consequence of Beauville’s theorem and the following “strict canonical class inequality”:
Theorem 2. Let $$f:S\to C$$ be a locally nontrivial semistable fibration of genus $$g\geq 2$$ with $$s$$ singular fibers. Then we have $$\deg f_* \omega_{S/C} <{g\over 2} (2g(C) -2+s)$$.
The validity of these two theorems is heavily dependent on the Miyaoka-Yau inequality.

##### MSC:
 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 14J17 Singularities of surfaces or higher-dimensional varieties 14H20 Singularities of curves, local rings