Global Invariants Of Trigonal Fibrations

To find out the relation among the invariants of an algebraic surface is an important problem in algebraic geometry. The computation of the invariants of fibred surfaces is equivalent to that of the relative invariants of the fibrations. There are effective methods to compute the invariants of surfaces with a ruling, elliptic fibration or hyperelliptic fibration. However, it is still an open problem to compute the invariants of a non-hyperelliptic fibration. Trigonal fibrations are the simplest cases of non-hyperelliptic fibrations. For example, genus 3 or 4 non-hyperelliptic fibrations are trigonal.Up to a base change, the surface of a trigonal fibration admits a generically triple cover over some ruled surface, so we can use the theory of triple covers to compute the invariants. We start from a triple cover over a geometric ruled surface, and use the canonical resolution to resolve the singularities of the covering surface. Then we have formulas for the computation of invariants of the smooth surface. In order to go back to our original surface, we have to contract the (—1)-curves in the fibers. Hence, the main problem is to count the number of vertical (—l)-curves coming from the canonical resolution. However, for triple covers, this is an open problem.For double covers, the similar problem is solved by E. Horikawa [19] and G. Xiao [61]. They divide the singular points of the branch locus into two types, and for each type, they have formulas to compute this number.For triple covers, we find a numerical classification of the singularities of the branch locus. We divide the singularities into 9 types, and for each type we know also the number of vertical (—1)-curves coming from the canonical resolution. As a consequence, we get the invariants of the original surface. In particular, if we apply the method to double covers, then we get also Horikawa and Xiao's result.As an application, we obtain the computation formulas for the invariants of a semi-stable non-hyperelliptic fibration of genus 3 based on the joint work of Z.-J. Chen and S.-L. Tan [12]. As a consequence, we solve M. Reid's conjecture [46] proposed in 1990. For a trigonal fibration of higher genus, we get upper bounds on its slope, which improves similar results of Z. Chen and S.-L. Tan.Furthermore, for n-tuple covers, we have a similar numerical classification of the singularities of branch locus.