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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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References:
[1] Arakelov, S.Ju: Families of algebraic curves with fixed degeneracy. Math. USSR Izv.5 (1971), 1277–1302 · Zbl 0248.14004 · doi:10.1070/IM1971v005n06ABEH001235
[2] Artin, M., Winters, G.: Degenerate fibres and stable reduction of curves. Topology10 (1971), 373–383 · Zbl 0221.14018 · doi:10.1016/0040-9383(71)90028-0
[3] Beauville, A.: Le nombre minimum de fibres singulières d’un courbe stable sur \(\mathbb{P}\)1 in Séminaire sur les pinceaux de courbes de genre au moins deux, ed. L. Szpiro Astérisque86 (1981), 97–108 · Zbl 0502.14009
[4] Beauville, A.: L’inégalitép g q pour les surfaces de type général. Bull. Soc. Math. France110 (1982), no. 3, 319–346
[5] Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0718.14023
[6] Chen, Z.: On the lower bound of the slope of a non-hyperelliptic fibration of genus 4. Intern. J. Math.4 (1993) no. 3, 367–378 · Zbl 0816.14006 · doi:10.1142/S0129167X93000194
[7] Deschamps, M.: Réduction semi-stable in Séminaire sur les pinceaux de courbes de genre au moins deux, ed. L. Szpiro Astérisque86 (1981), 1–34
[8] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. IHES36 (1969), 75–109 · Zbl 0181.48803 · doi:10.1007/BF02684599
[9] Esnault, H., Viehweg, E.: Effective bounds for semipositive sheaves and the height of points on curves over complex function fields. Compositio Mathematica76 (1990), 69–85 · Zbl 0742.14020
[10] Hirzebruch, F.: Singularities of algebraic surfaces and characteristics numbers. The Lefschetz Centennial Conference, Part I (Mexico City) Contemp. Math.58 (1986), Amer. Math. Soc. Providence, R.I. 141–155
[11] Horikawa, E.: Notes on canonical surfaces. Tohoku Math. J.43 (1991), 141–148 · Zbl 0748.14014 · doi:10.2748/tmj/1178227542
[12] Konno, K.: Non-hyperelliptic fibrations of small genus and certain irregular canonical surfaces. Preprint (1992) · Zbl 0822.14009
[13] Miyaoka, Y.: The maximal number of quotient singularities on surfaces with given numerical invariants. Math. Ann.268, 159–171 · Zbl 0521.14013
[14] Parshin, A.N.: Algebraic curves over function fields I. Math. USSR Izv.2 (1968), 1145–1170 · Zbl 0188.53003 · doi:10.1070/IM1968v002n05ABEH000723
[15] Reid, M.: Problems on pencils of small genus. Preprint
[16] Szpiro, L.: Propriété numériques de faisceau dualisant relatif in Séminaire sur les pinceaux de courbes de genre au moins deux. ed. L. Szpiro Astérisque86 (1981), 44–78
[17] Tan, S.-L.: On the invariants of base changes of pencils of curves, I. Manuscripta Math.84 (1994), 225–244 · Zbl 0835.14012 · doi:10.1007/BF02567455
[18] Tan, S.-L.: The minimal number of singular fibers of a semistable curve over £1. J. Algebraic Geometry4 (1995), 591–596 · Zbl 0864.14003
[19] Vojta, P.: Diophantine inequalities and Arakelov theory in Lang, S., Introduction to Arakelov Theory (1988), Springer-Verlag, 155–178
[20] Xiao, G.: Fibered algebraic surfaces with low slope. Math. Ann.276 (1987), 449–466 · Zbl 0596.14028 · doi:10.1007/BF01450841
[21] Xiao, G.: Problem list in: Birational geometry of algebraic varieties: open problems. The 23rd International Symposium of the Taniguchi Foundation, (1988), pp. 36–41
[22] Xiao, G.: On the stable reduction of pencils of curves. Math. Z.203 (1990), 379–389 · Zbl 0662.14013 · doi:10.1007/BF02570745
[23] Xiao, G.: The fibrations of algebraic surfaces. Shanghai Scientific & Technical Publishers 1992. (Chinese)
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