F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections

We study F-theory compactifications with U(1)×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 E with two rational points and a zero point is the generic Calabi-Yau onefold in dP2. We determine the birational map to its Tate and Weierstrass 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 = P2. A thorough analysis of the generic codimension two singularities of these elliptic Calabi-Yau manifolds is presented. This determines the general U(1)×U(1)-charges of matter in corresponding F-theory compactifications. The matter multiplici-ties for the fibration over P2 are determined explicitly and shown to be consistent with anomaly cancellation. Explicit toric examples are constructed, both with U(1)×U(1) and SU(5)×U(1)×U(1) gauge symmetry. As a by-product, we prove the birational equiva-lence of the two elliptic fibrations with elliptic fibers in the two blow-ups Bl(1,0,0)P²(1, 2, 3) and Bl(0,1,0)P2(1, 1, 2) employing birational maps and extremal transitions