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We introduce a new model for elliptic fibrations endowed with a Mordell-Weilgroup of rank one. We call it a Q$_7(\mathscr{L},\mathscr{S})$ model.  It naturally generalizes several previous models of elliptic fibrationspopular in the F-theory literature. The model is also explicitly smooth, thusrelevant physical quantities can be computed in terms of topological invariantsin straight manner. Since the general fiber is defined by a cubic curve, basicarithmetic operations on the curve can be done using the chord-tangent grouplaw. We will use this model to determine the spectrum of singular fibers of anelliptic fibration of rank one and compute a generating function for its Eulercharacteristic. With a view toward string theory, we determine a semi-stabledegeneration which is understood as a weak coupling limit in F-theory. We showthat it satisfies a non-trivial topological relation at the level ofhomological Chern classes. This relation ensures that the D3 charge in F-theoryis the same as the one in the weak coupling limit.

A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations