Towards the Standard Model spectrum from elliptic Calabi-Yau

We show that it is possible to construct supersymmetric three-generation models of Standard Model gauge group in the framework of non-simply-connected elliptically fibered Calabi-Yau, without section but with a bi-section. The fibrations on a cover Calabi-Yau, where the model has 6 generations of SU(5) and the bundle is given via the spectral cover description, use a different description of the elliptic curve which leads to more than one global section. We present two examples: one cover Calabi-Yau is a fibre product of rational elliptic surfaces $dP_9$ with conventional chiral matter; another example is an elliptic fibration over a Hirzebruch surface, where the bundles turn on a further parameter, related to singularities of the fibration and the branching of the spectral cover, to get chiral matter.