On the invariants of base changes of pencils of curves. I
Let \(f : S \to C\) be a relatively minimal fibration of a smooth projective surface \(S\) over a curve \(C\). Denote by \(g\) the genus of a general fiber of \(f\), by \(K_{S/C}\) the relative canonical divisor, \(\chi_f : = \deg f_* K_{S/C}\), \(K^2_f : = K^2_{S/C}\) and \(e_f : = \sum_{p \in C} (e(f^{- 1} (p)) - e)\) the relative Euler characteristic. The well-known Parshin-Arakelov theorem asserts that \(\chi_f \geq 0\), \(K^2_f \geq 0\) and if \(g > 1\) one of them is 0 iff the fibration is locally trivial. Furthermore, it is known that \(e_f \geq 0\) and, if \(g > 1\), \(e_f = 0\) iff \(f\) is smooth. Let \(\pi : \widetilde C \to C\) be a base change of degree \(d\). The pull-back fibration \(\widetilde f : \widetilde S \to \widetilde C\) is, by definition, obtained from the minimal desingularization \(S_2\) of the normalization \(S_1\) of \(S\times_C \widetilde C\) by the contraction of the \((- 1)\)-curves contained in the fibers of \(S_2/ \widetilde C\). Put \(\chi_\pi : = d \chi_f - \chi_{\widetilde f}\), \(K_\pi : = dK^2_f - K^2_{\widetilde f}\), \(e_\pi : = de_f - e_{\widetilde f}\). \textit{G. Xiao} has proved that \(\chi_\pi \geq 0\), \(K^2_\pi \geq 0\) and, if \(g > 1\), one of them is 0 iff \(\pi\) is an ``invariant base change'' with respect to \(f\). He also conjectured that the same should be true for \(e_\pi\). It is the aim of the present paper to prove this conjecture. The proof is based on some new inequalities between the invariants of the singularities of \(z^d = f(x,y)\). The author also recalls the proof of Xiao's result which is originally published in Chinese.
scientific article; zbMATH DE number 703882
scientific article