---
_id: '66329'
abstract:
- lang: eng
  text: 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.
author:
- first_name: Madeline
  full_name: Brandt, Madeline
  last_name: Brandt
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: 'Brandt M, Ulirsch M. Symmetric powers of algebraic and tropical curves: a
    non-Archimedean perspective. <i>arXiv:181208740</i>. Published online 2018. doi:<a
    href="https://doi.org/10.1090/btran/113">10.1090/btran/113</a>'
  apa: 'Brandt, M., &#38; Ulirsch, M. (2018). Symmetric powers of algebraic and tropical
    curves: a non-Archimedean perspective. <i>ArXiv:1812.08740</i>. <a href="https://doi.org/10.1090/btran/113">https://doi.org/10.1090/btran/113</a>'
  bibtex: '@article{Brandt_Ulirsch_2018, title={Symmetric powers of algebraic and
    tropical curves: a non-Archimedean perspective}, DOI={<a href="https://doi.org/10.1090/btran/113">10.1090/btran/113</a>},
    journal={arXiv:1812.08740}, author={Brandt, Madeline and Ulirsch, Martin}, year={2018}
    }'
  chicago: 'Brandt, Madeline, and Martin Ulirsch. “Symmetric Powers of Algebraic and
    Tropical Curves: A Non-Archimedean Perspective.” <i>ArXiv:1812.08740</i>, 2018.
    <a href="https://doi.org/10.1090/btran/113">https://doi.org/10.1090/btran/113</a>.'
  ieee: 'M. Brandt and M. Ulirsch, “Symmetric powers of algebraic and tropical curves:
    a non-Archimedean perspective,” <i>arXiv:1812.08740</i>, 2018, doi: <a href="https://doi.org/10.1090/btran/113">10.1090/btran/113</a>.'
  mla: 'Brandt, Madeline, and Martin Ulirsch. “Symmetric Powers of Algebraic and Tropical
    Curves: A Non-Archimedean Perspective.” <i>ArXiv:1812.08740</i>, 2018, doi:<a
    href="https://doi.org/10.1090/btran/113">10.1090/btran/113</a>.'
  short: M. Brandt, M. Ulirsch, ArXiv:1812.08740 (2018).
date_created: 2026-07-08T06:52:13Z
date_updated: 2026-07-08T09:45:46Z
doi: 10.1090/btran/113
external_id:
  arxiv:
  - '1812.08740'
language:
- iso: eng
publication: arXiv:1812.08740
status: public
title: 'Symmetric powers of algebraic and tropical curves: a non-Archimedean perspective'
type: journal_article
user_id: '82981'
year: '2018'
...