  In this paper, we study the deformation theory of degenerate algebraic curves
on singular varieties which appear as the degenerate limit of families of
varieties. For this purpose, we systematically develop a new method to
calculate the obstruction cohomology class of degenerate algebraic curves. This
enables us to judge whether a given degenerate curve can be deformed to a
smooth curve or not in variety of situations. In this paper, we apply it to
curves of genus one on degeneration of toric varieties. In particular, we
obtain the necessary and sufficient condition for the realizability of tropical
curves of genus one, extending various results obtained so far.
