On moduli spaces in algebraic and tropical geometry - Taboada, Danny Ariel Flores (2023)
We introduce and study polystable divisors on a tropical curve, which are the tropical analogue of polystable torsion-free rank-1 sheaves on a nodal curve. We construct a universal tropical Jacobian over the moduli space of tropical curves of genus g. This space parametrizes equivalence classes of tropical curves of genus g together with a µ-polystable divisor, and can be seen as a tropical counterpart of Caporaso universal Picard scheme. We describe polyhedral decompositions of the Jacobian of a tropical curve via polystable divisors, relating them with other known polyhedral decompositions. We also study a compactification of the moduli space of theta characteristics and give a modular interpretation of the geometric points, and describe the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves, and describe explicitly the induced decomposition of its cones.