F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections

We study F-theory compactifications with U(1) x U(1) gauge symmetry on elliptically fibered Calabi-Yau manifolds with a rank two Mordell-Weil group. We find that the natural presentation of an elliptic curve epsilon with two rational points and a zero point is the generic Calabi-Yau onefold in dP(2). We determine the birational map to its Tate and Weier-strass form and the coordinates of the two rational points in Weierstrass form. We discuss its resolved elliptic fibrations over a general base B and classify them in the case of B = P-2. A thorough analysis of the generic codintension two singularities of these elliptic Calabi-Yau manifolds is presented. This determines the general U(1) x U(1)-charges of matter in corresponding F-theory compactifications. The matter multiplicities for the fibration over P-2 are determined explicitly and shown to be consistent with anomaly cancellation. Explicit tonic examples are constructed, both with U(1) x U(1) and SU(5) x U(1) x U(1) gauge symmetry. As a by-product, we prove the birational equivalence of the two elliptic fibrations with elliptic fibers in the two blow-ups Bl((1,0,0))P(2)(1,2,3) and Bl((0,1,0))P(2)(1,1,2) employing birational maps and extremal transitions.