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

Madeline Brandt; Martin Ulirsch

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arXiv · 1812.08740

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

Madeline Brandt·Martin Ulirsch

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Abstract

We show that the non-Archimedean skeleton of the $d$-th symmetric power of a smooth projective algebraic curve $X$ is naturally isomorphic to the $d$-th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of $X$. 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.

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Madeline Brandt, Martin Ulirsch. 2018-12-20. Symmetric powers of algebraic and tropical curves: a non-Archimedean perspective. https://arxiv.org/abs/1812.08740

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