On moduli spaces in algebraic and tropical geometry

• Ano de defesa: 2023
• Autor(a) principal: Taboada, Danny Ariel Flores
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Palavras-chave em Português: algebraic curve; moduli space; Spin curve; Universal tropical Jacobian; Tropical curve; Álgebra; Curva algébrica; Curva matemática; Curva tropical; Espaço de moduli; Curva spin; Jacobiana universal tropical
• Link de acesso: http://app.uff.br/riuff/handle/1/29221

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.