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On the invariants of base changes of pencils of curves. II. (English) Zbl 0864.14016
[For part I of this paper see Manuscr. Math. 84, No. 3-4, 225-244 (1994; Zbl 0835.14012).]
Let \(f:S\to C\) be a pencil of curves of genus \(g\geq 2\). We assume that \(S\) contains no \((-1)\)-curves in a fiber of \(f\), and denote by \(K^2_{S/C}\), \(\chi_f= \deg f_* \omega_{S/C}\), \(e_f= \sum_F(\chi_{\text{top}} (F)-(2-2g))\) the standard relative invariants of \(f\). In this paper, we associate to each (singular) fiber \(F\) of \(f\) three nonnegative rational numbers \(c^2_1(F)\), \(c_2(F)\) and \(\chi_F\), which measure the semistability of \(F\), e.g., one of them vanishes iff \(F\) is semistable. They satisfy the Noether equality, \(c^2_1(F)\leq 2c_2(F)\) and \(c^2_1(F) \leq 4g-4\). They can be computed directly from the embedded resolution of the singularities of \(F_{\text{red}}\). We define \[ I_K(f) =K^2_{S/C}- \sum_Fc^2_1(F), \quad I_\chi (f)= \chi_f- \sum_F\chi_F, \quad I_e(f) =e_f- \sum_Fc_2(F), \] where \(F\) runs over the singular fibers of \(f\). These three numbers are nonnegative invariants of \(f\), and if \(f\) is semistable, then they are nothing but the relative invariants of \(f\).
The main result of this paper is: \(\pi:\widetilde C\to C\) is a finite base change of degree \(d\), and \(\widetilde f: \widetilde S\to \widetilde C\) the pullback fibration of \(f\) under \(\pi\), i.e., the unique relative minimal model of \(S\times_C\widetilde C\to \widetilde C\), then \[ I_K(\widetilde f) =dI_K(f), \quad I_\chi (\widetilde f)= dI_\chi(f), \quad I_e(\widetilde f)= dI_e(f). \] This result has many interesting applications. First, it gives a numerical criterion for isotriviality: \(f\) is isotrivial (i.e., all the nonsingular fibers of \(f\) are isomorphic) iff \(I_K(f)=0\), or equivalently \(I\chi (f)=0\). It gives also a numerical criterion for a base change to be stabilizing. Then, we obtain a canonical class inequality for any fibration, which generalizes Vojta’s inequality to the non-semistable case. As the third application, the theorem says that the slops (i.e., \(K^2/ \chi)\) of the semistable models of \(f\) are independent of the base changes, and determined uniquely by the relative invariants of \(f\) and the local invariants of the singular fibers. This solves a problem of Xiao [problem 7 in “Birational geometry of algebraic varieties: open problems”, The 23rd International Symposium of the Taniguchi Foundation (1988)]. Finally, we give a formula to compute the Horikawa numbers of non-semistable singular fibers in a non-hyperelliptic pencil of curves of genus 3.
If the pencil \(f\) (not necessarily semistable) is not isotrivial, then it defines a finite morphism of \(C\) to the moduli space of stable curves of genus \(g\). Then there are three well-known invariants of \(f\) which are defined as the degrees on \(C\) of the pullback classes of the Hodge class \(\lambda\), the boundary class \(\delta\) and the class \(\kappa\). From our main result, these three invariants are just the invariants \(I_\chi(f)\), \(I_e(f)\) and \(I_K(f)\). If \(f\) is semistable, this relationship is well-known.
Reviewer: S.Tan (Shanghai)

14H10 Families, moduli of curves (algebraic)
14J17 Singularities of surfaces or higher-dimensional varieties
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