Tropicalization of theta characteristics, double covers, and Prym varieties

We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of $$2^{g-1}$$ even theta characteristics and $$2^{g-1}$$ odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.