Obstructions associated with tropical curves

Given a complex variety $X$, suppose there is a degeneration $\mathfrak{X}$ of it to a union of toric varieties glued along toric divisors. Then, one can use tropical curves to study holomorphic curves on $X$ through degenerate curves on the central fiber $X_{0}$ of $\mathfrak{X}$. In such studies, the case corresponding to so-called non-superabundant tropical curves is relatively well understood. This is the case where the curve satisfies a version of transversality. On the other hand, in the case corresponding to superabundant tropical curves, where transversality fails, not much is known. The first important step for such a study is describing the obstruction to deforming degenerate curves on $X_{0}$ to generic fibers of $\mathfrak{X}$. In this paper, we present a general formula describing such an obstruction. In the study of superabundant tropical curves, allowing higher-valent vertices (i.e., those with valency greater than three) is essential, although there have been few studies of them in this context. Our formula covers such cases.