  We establish the existence of complete K\"ahler metrics of semi-positive
holomorphic sectional curvature with many zeroes in an interesting and natural
geometric setting. Specifically, we use Calabi's Ansatz in the form due to
Koiso-Sakane to produce such metrics on the total space of powers of the
tautological line bundles over the projective line. We formulate a general
conjecture regarding the compact case and use a product approach to obtain an
analogous result in the non-compact case.
