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On extremal rational elliptic surfaces. (English) Zbl 0652.14003
We classify all elliptic fibrations on rational elliptic surfaces with only a finite group of sections. In view of the Shioda-Tate formula and the automatic Picard number 10 of rational surfaces, this is equivalent to fibrations which are maximally reducible. (The sum of the ranks of all reducible fibers equals $$8;rank=(\#components-1))$$. We call such fibrations extremal. It turns out to be 16 different cases (one of which depending on moduli) and the semistable ones (only multiplicative fibers) are classified by fourtuples of integers summing up to 12 (the Euler characteristic) and whose products are squares.
The techniques used are both combinatorial (the combinatorial data of the J-map) and geometrical exhibiting actual constructions. In subsequent papers we have classified all possible configurations of singular fibers on rational surfaces (around 300 cases) and all possible semistable configurations on K3 surfaces (around 1200). The classification of all extremal fibrations on a K3 surface (the next case) might turn out to be a severe test of human endurance.