Symmetric powers of algebraic and tropical curves: A non-Archimedean perspective

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We show that the non-Archimedean skeleton of the d d -th symmetric power of a smooth projective algebraic curve X X is naturally isomorphic to the d d -th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of X X . The retraction to the skeleton is precisely the specialization map for divisors. Moreover, we show that the process of tropicalization naturally commutes with the diagonal morphisms and the Abel-Jacobi map and we exhibit a faithful tropicalization for symmetric powers of curves. Finally, we prove a version of the Bieri-Groves Theorem that allows us, under certain tropical genericity assumptions, to deduce a new tropical Riemann-Roch-Theorem for the tropicalization of linear systems.