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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.

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