  This is the first of a set of papers having the aim to provide a detailed
description of brane configurations on a family of noncompact threedimensional
Calabi-Yau manifolds. The starting point is the singular manifold C^3/Z_6,
which admits five distinct crepant resolutions. Here we apply local mirror
symmetry to partially determine the prepotential encoding the GW-invariants of
the resolved varieties. It results that such prepotential provides all numbers
but the ones corresponding to curves having null intersection with the compact
divisor. This is realized by means of a conjecture, due to S. Hosono, so that
our results provide a check confirming at least in part the conjecture.
