On Isotrivial Belyi Fibrations Of Small Genus

An important tool to classify algebraic surfaces is the fibrations. From the geometric point of view, each meromorphic function can be regarded as a fibration over P1 (not necessarily fibre-connected). One of the basic problems is to study properties of fibrations over P1 with connected fibers. From the results of [Beal] and [Tan2], the minimum number of singular fibers (i.e. the minimum of critical points of a meromorphic function) is 2 for a nontrivial fibration. In case of non-isotrivial (resp. semistable) fibration the minimum number is 3 (resp.5).We call a fibration f over P1 with 2 or 3 singular fibers, a Belyi fibration. This is because of an important property of algebraic curves, namely a curve defined over a number field is isomorphic to a complex curve admitting a finite cover over P1 with 2 or 3 branch points (Belyi’s Theorem). Belyi fibrations with 2 singular fibers have already been classified in case of genus 1 and 2 by U. Schmickler Hirzebruch ([Hir]) and by C. Gong, J. Lu, S.-L. Tan ([GLT1]) respectively.In this thesis, we will classify isotrivial, relatively minimal Belyi fibrations with 3 singular fibers in case of genus 1 and genus 2. For genus 1, we get 13 such families up to isomorphism. For genus 2, number of families are 14 up to twisting of fibers by t or t - 1 or both. In both cases, we list them explicitly and find the equations of each of these families. Since in this thesis, we assume that the fibrations are isotrivial, so their semistable model is smooth or in geometric language, the topology monodromy of singular fibers is periodic.The method of solving genus 1 is interesting. We see from the Kodaira’s list of singular fibers (see Appendix 6.1), there are 7 out of 11 singular fibers which are pe-riodic. The first step is to find the possible combinations of the three singular fibers numerically out of these 7, i.e. by using Lu and Tan’s results [LT] on inequalities on the chern numbers of these singular fibers. Since, we have elliptic curves as our singular fibers, so we use its J-invariant to find the equations of the families. Since by assumption, the fibrations are isotrivial, so the J-invariant is constant.The solution of genus 2 is a bit tricky. First of all, we get 17 fibers which are periodic out of total 126, which are completely classified by Namikawa and Ueno (see [NU]). In the first step, when we find all the possible combinations of three out of these 17, we get 151 combinations. Then we use a lemma by Ishizaka [Ish2] to find the fiber twist and reduce our problem of finding the equations of these fami-lies from 151 to 50. We use double covers, a technique used in [GLT1] to find the equations of the families. While finding the equations, we rule out several families which do not exist, because of their contradiction with the local equations of their branch curves. Finally, we are left with just 14 families that exist (up to fiber twist).