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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)

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