Semistable genus 5 general type \(\mathbb P^1\)-curves have at least 7 singular fibres (Q2856406)

This paper concerns \(\mathbb P^1\)--curves, i.e. fibrations \(f: \, X \longrightarrow \mathbb P^1\) whose generic fiber is a curve and \(X\) is a surface of general type.NEWLINENEWLINEWe say that \(f\) is \textit{relatively minimal} if there is no \((-1)\)--curve contained in any of its fibers. Moreover, we say that \(f\) is \textit{semistable} if in addition all the singular fibers are reduced nodal curves. Finally, \(f\) is called \textit{isotrivial} if all smooth fibers are mutually isomorphic (see [\textit{E. Sernesi}, ``General curves on algebraic surfaces'', accepted in J. Reine u. Angew. Math., \url{doi:10.1515/crelle-2013-0019}, \url{arXiv:math/0702865}] for an overview of these notions).NEWLINENEWLINEA classical problem is to determine a lower bound for the number \(s\) of singular fibers of \(f\), according to the properties fulfilled by \(f\). In this framework, an important initial result is proved by \textit{A. Beauville} [Astérisque 86, 97--108 (1981; Zbl 0502.14009)]: if \(f\) is non--isotrivial then \(s \geq 3\) and, if \(f\) is semistable, then \(s \geq 4\).NEWLINENEWLINELater results establish a lower bound for \(s\) related with the genus \(g\) of the generic fiber of \(f\). Still Beauville in [\textit{A. Beauville}, C.R. Acad. Sci. Paris, Sér. I 294, 657--660 (1982; Zbl 0504.14016)] gives a classification of semistable fibrations where \(g=1\) and \(s=4\).NEWLINENEWLINELet us mention also [\textit{S.-L. Tan}, J. Algebr. Geom. 4, N. 3, 591--596 (1995; Zbl 0864.14003)], where it is proved that if \(f\) is semistable and \(g \geq 2\), then \(s \geq 5\). So a natural problem is to classify semistable fibrations having \(g \geq 2\) and \(s=5\) or \(6\). Within this context, in [\textit{S.-L. Tan, Y. Tu} and \textit{A. G. Zamora}, Math. Z. 249, No. 2, 427--438 (2005; Zbl 1074.14009)] the authors prove that, if \(f\) is relatively minimal then:NEWLINENEWLINE(1) \(s=5\) implies that \(X\) is birationally rational or ruled;NEWLINENEWLINE(2) \(s=6\) and \(g=2,3,4\) imply that \(X\) is not of general type;NEWLINENEWLINE(3) \(s=6\), \(g=5\) and \(X\) of general type imply that the minimal model \(S\) of \(X\) satisfies NEWLINE\[NEWLINE K^2_S=1, \quad p_g(S) =2, \quad q(S) =0 NEWLINE\]NEWLINE and in this last case \(f\) comes from a pencil of genus \(5\) curves with \(5\) base points.NEWLINENEWLINESince the authors have no example of case \((3)\), they conjecture thatNEWLINENEWLINE\centerline \textit{if \(X\) is of general type and \(g=5\) then \(s \geq 7\).}NEWLINENEWLINEHence, in order to prove this, it suffices to show that a surface \(S\) of general type such that \(K^2_S=1, \; p_g(S) =2, \; q(S) =0\) does not admit of a pencil of genus \(5\) curves with \(5\) base points.NEWLINENEWLINEThis fact is exactly the purpose of the current paper: namely the authors show in Theorem 1 that such pencil does not exist by lifting the bicanonical map of \(S\) (as double covering of a quadric cone) to a map from a suitable blowing up \(\overline S\) of \(S\) to the desingularization \(\mathbb F^2\) of the cone. An accurate study of the lifted pencils of curves on the surface \(\overline S\), and also on the Hirzebruch surface \(\mathbb F^2\), leads to the desired result. Therefore also the conjecture of Tan-Tu-Zamora is proved.NEWLINENEWLINEThe paper is very short and maybe poor in background of notions and definitions needed to understand the problem. Nevertheless the result is interesting and placed in the wide framework of problems concerning fibrations.