I want to sketch the existence of Q-split models of rational elliptic surfaces for every "P.O.S-type" (i.e. with type of singular fibres classified by Persson and with the structure type of Mordell-Wei lattices given by Oguiso-Shioda). By a Q-split model of such, I mean an elliptic curve E/Q(t) for which the Q(t)-rational points give the full MWL, while the reducible singular fibres of the associated elliptic surface have all the irreducible components defined over Q. The idear of proof is to use "excellent families" and analog of "vanishing cycles". |

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