×

zbMATH — the first resource for mathematics

On the invariants of base changes of pencils of curves. I. (English) Zbl 0835.14012
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 \(\chi_\pi : = d \chi_f - \chi_{\widetilde f}\), \(K_\pi : = dK^2_f - K^2_{\widetilde f}\), \(e_\pi : = de_f - e_{\widetilde f}\). 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.

MSC:
14J17 Singularities of surfaces or higher-dimensional varieties
14H20 Singularities of curves, local rings
14D99 Families, fibrations in algebraic geometry
14J26 Rational and ruled surfaces
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [Ar] Arakelov, S. Ju.,Families of algebraic curves with fixed degeneracy, Math. USSR Izv.5 (1971), 1277–1302 · Zbl 0248.14004
[2] [AW] Artin, M., Winters, G.,Degenerate fibres and stable reduction of curves, Topology10 (1971), 373–383 · Zbl 0221.14018
[3] [As] Ashikaga, T.,Normal two-dimensional hypersurface triple points and Horikawa type resolution, Tôhoku Math. J.44 (1992), 177–200. · Zbl 0801.14011
[4] [BPV] Barth, W., Peters, C., Van de Ven, A.,Compact Complex Surfaces, Berlin, Heidelberg, New York: Springer, 1984 · Zbl 0718.14023
[5] [Be] Beauville, A.,L’inégualité p g q-4 pour les surfaces de type général, Bull. Sco. Math. France110 (1982), no. 3, 343–346
[6] [DM] Deligne, P., Mumford, D.,The irreducibility of the space of curves of given genus, Publ. IHES36 (1969), 75–109 · Zbl 0181.48803
[7] [Du] Durfee, A. H.,The signature of smoothings of complex surface singularities, Math. Ann.232 (1978), 85–98 · Zbl 0357.32008
[8] [Ha] Hartshorne, R.,Algebraic Geometry, GTM 52, Springer-Verlag, 1977
[9] [Ho] Horikawa, E.,On deformations of quintic surfaces, Inv. Math.31 (1975), 43–85 · Zbl 0317.14018
[10] [HR] Hauser, H., Randell, R.,Report on the problem session, Singularities, (R. Randell, eds.), Contemp. Math. vol.90 (1989), pp. 119–134 · Zbl 0673.32001
[11] [La] Laufer, H. B.,On {\(\mu\)} for surface singularities, Several Complex Variables, Part I (Wells, R. O., eds.), Proc. of Symposia in Pure Math., vol.30 Providence, Rhode Island: Amer. Math. Soc. (1977), pp. 45–49
[12] [Mi] Milnor, J.,Singular points of complex hypersurfaces, Ann. Math. Studies, vol.61, Princeton University Press, Princeton, N. J., 1968 · Zbl 0184.48405
[13] [Pa] Parshin, A. N.,Algebraic curves over function fields I, Math. USSR Izv.2 (1968), 1145–1170 · Zbl 0188.53003
[14] [To] Tomari, M.,The inequality 8p g <{\(\mu\)} for hypersurface two-dimensional isolated double points, Math. Nachr.164 (1993), 37–48 · Zbl 0799.32032
[15] [X1] Xiao, G.,Problem list, In: Birational geometry of algebraic varieties: open problems. The 23rd International Symposium of the Taniguchi Foundation, (1988), pp. 36–41
[16] [X2] Xiao, G.,On the stable reduction of pencils of curves, Math. Z.203 (1990), 379–389 · Zbl 0662.14013
[17] [X3] Xiao, G.,The fibrations of algebraic surfaces, Shanghai Scientific & Technical Publishers, 1992. (Chinese)
[18] [XY] Xu, Y.-J., Yau, S. S.-T.,A sharp estimate of the number of integral points in a tetrahedron, J. reine angew. Math.423 (1992), 199–219
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.