Abstract: Let X be a scheme equipped with a SNC divisor D. I will sketch the construction of a proper moduli space of subschemes Z in expansions of X such that Z satisfies an appropriate transversality condition to D. The central insight is a tropical understanding for when Z is flat over the Artin fan of X, and how this behaves in families. I will present the example of the logarithmic linear system, a toric modification of projective space. The associated fan is closely related to the geometry of tropical curves and secondary polytopes. Joint work with Dhruv Ranganathan.