  Spaces of holomorphic maps from the Riemann sphere to various complex
manifolds (holomorphic curves ) have played an important role in several area
of mathematics. In a seminal paper G. Segal investigated the homotopy type of
holomorphic curves on complex projective spaces and M. Guest on compact smooth
toric varieties.. Recently Mostovoy and Villanueva, obtained a far reaching
generalisation of these results, and in particular (for holomorphic curves)
improved the stability dimension obtained by Guest. In this paper, we
generalize their result to holomorphic curves, on certain non-compact smooth
toric varieties.
