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

D-Branes on C^3_6 part I: prepotential and GW-invariants