  We analyze heterotic line bundle models on elliptically fibered Calabi-Yau
three-folds over weak Fano bases. In order to facilitate Wilson line breaking
to the standard model group, we focus on elliptically fibered three-folds with
a second section and a freely-acting involution. Specifically, we consider
toric weak Fano surfaces as base manifolds and identify six such manifolds with
the required properties. The requisite mathematical tools for the construction
of line bundle models on these spaces, including the calculation of line bundle
cohomology, are developed. A computer scan leads to more than $400$ line bundle
models with the right number of families and an $SU(5)$ GUT group which can
descend to standard-like models after taking the $\mathbb{Z}_2$ quotient. A
common and surprising feature of these models is the presence of a large number
of vector-like states.
