The Minimal Number Of Singular Fibers Of A Semi-stable Pencil

Fibrations of algebraic varieties are important subject in algebraic geometry. On the one hand,it is a method to reduce the dimension of algebraic varieties,namely to understand the global properties of the total space by the local properties of the fibers.On the other hand,it provides the geometric analogues of some problems in number theory,namely,the geometric analogues of number fields are algebraic function fields.The main purpose of this paper is to study the number s of singular fibers of a surface fibration f:S→P~1 over rational curve.This is called Szpiro Problem,and the first progress on this problem was made by Beauville in 1980.In fact,the problem has a closed relationship with Arakelov's inequality.Szpiro wanted to know the minimal number s of the critical points[27]. Beauville showed that s≥4,and constructed examples of s=6 for arbitrary genus g≥2.He conjectured that if g≥2,then s≥5[6].Using the canonical class inequality,Sheng-Li Tan proves Beauville's conjecture[28].We will prove that s≥6 when g≥18,with only one possible exceptional case which is a very simple double cover of C×P~1.The number of critical points depends also on the complexity of the surface S.[29]proved that if the Kodaira dimension of S is non-negative, then s≥6,and conjectured that s≥7 if S is of general type.We will confirm this conjecture for g≥51.The equality in Arakelov's inequality implies a certain number theoretic property.The semi-stable pencils of elliptic curves with s=4 can be divided into six modular families of elliptic curves[7].Using the results on the adjoint linear system of a fiber in the second chapter,we will show that a semi-stable pencil of curves with exactly 5 critical points must be of genus 2 provided its Neron model admits only 4 critical points.Reider's method is very powerful in the study of the adjoint linear system |K_S+L| of a positive line bundler L[25],which greatly simplifies Bombieri's work on pluri-canonical maps of surfaces.Mendes lops deals with the case when L is semi-positive and 1-connected based on Francia's work[21][12]. Applying their results in our situation,we will prove that the singular base points of the adjoint linear system |K_S+F|are exactly the separated double points of F.Using the canonical mapping,Horikawa classified the surfaces of general type on the Noether line K_S~2=2P_g-4[15].Based on the Miyaoka-Yau's inequality,Beauville made a profound study on the behavior of the canonical map[5].Kitagawa and Konno use the map of |K_S+F| to study the fibration on a rational surface X and get some properties of Mordell-Weil lattice[18]. Similarly,I will classify the fibrations on a rational surfaces with small relative invariants by using the map of |K_S+F|.Theorems on finiteness are closely related to the hyperbolicity of the region of non-critical point.The canonical class inequality derived from Miyaoka-Yau inequality or Schwarz-Yau Lemma controls the bending of the moduli space of curves[31][28][16],the Arakelov inequality derived from variation of Hodge structure controls the bending of the moduli space of Abelian varieties[32].In the third chapter we will study three relevant problem. In particular,we obtain a formula to compute the relative holomorphic Euler characteristic by using an elementary method,and we have a new proof of the Arakelov inequality.