Note on three-generation models in heterotic string and F-theory on elliptic Calabi-Yau manifolds over Hirzebruch varieties

We give a complete list of a class of three-generation models in E8 x E8 heterotic string theory and its dual F-theory on an elliptic Calabi-Yau over a (generalized) Hirzebruch variety in which the divisors of the relevant line bundles needed for a smooth Weierstrass model are effective. The most stringent constraint on the bound of the eta class comes from the effectiveness of the divisor of the bundle corresponding to the highest Casimir in Looijenga's weighted projective space, as well as from the compactness of the toric variety. Comparison is also made with the list obtained in the literature.