Fibration Of Genus3with Two Singular Fibers Over P~1

The fibrations of algebraic surfaces play an important role in the classification of surfaces. It is well known that each meromorphic function can be regarded as a fibration over P1(not necessary fibre-connected) from geometric point of view. One of the basic problems is to study the property of fibration over P1with connected fibers. Prom the results in [Bea1] and [Tan2], the number of singular fibers (i.e. the number of critical points of the meromorphic function) is at least2. Moreover, if this fibration is also non-isotrivial (resp., semistable), then the number of singular fibers is at least3(resp., at least5).The fibrations over P1with2singular fibres were studied by many people. For example, U. Schmickler Hirzebruch ([Hir]) claimed that there are exactly5families of elliptic fibrations with2singular fibres which are dual to each other in the notion of Kodaira.[GLT2, GLT4] investigate such fibrations by a new h1,1inequality technique and give the completely classification of such fibrations of genus2together with the Mordell-Weil lattices.In this paper we will discuss the properties of the fibration of genus3over P1with2singular fibres. We will solve the following two questions:(1) Give the complete classification of such fibration.(2) Compute their Mordell-Weil lattices precisely.By the famous result of S-L.Tan and Y-P.Tu and Zamora [TTZ], every fibration of genus g over P1satisfy the inequality Kf2≥4g-4, when the equality holds the fibration can be realized as a double cover over a ruled surface. In the case that g=3and Kf2=8, Problem (1) still can be deal with by double covering techniques. The harder part is to investigate the case Kf2>8especially when f is a nonhyperelliptic fibration of genus3. To solve this problem completely, we use the technique of triple covering and canonical resolution of singular points.By using the technique from [GLT3] we can solve problem (2) partly, however, that technique won’t work out in some cases. This is another harder parts that we need to solve. From the complete classification equation of problem (1), one can determine the Mordell-Weil lattices in the remained case precisely. Thus the problem (2) be answered perfectly.