---
_id: '66322'
abstract:
- lang: eng
  text: 'Motivated by the realizability problem for principal tropical divisors with
    a fixed ramification profile, we explore the tropical geometry of the double ramification
    locus in $\mathcal{M}_{g,n}$.There are two ways to define a tropical analogue
    of the double ramification locus: one as a locus of principal divisors, the other
    as a locus of finite effective ramified covers of a tree. We show that both loci
    admit a structure of a generalized cone complex in $M_{g,n}^{trop}$, with the
    latter contained in the former. We prove that the locus of principal divisors
    has cones of codimension zero in $M_{g,n}^{trop}$, while the locus of ramified
    covers has the expected codimension $g$. This solves the deformation-theoretic
    part of the realizability problem for principal divisors, reducing it to the so-called
    Hurwitz existence problem for covers of a fixed ramification type.'
author:
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
- first_name: Dmitry
  full_name: Zakharov, Dmitry
  last_name: Zakharov
citation:
  ama: Ulirsch M, Zakharov D. Tropical double ramification loci. <i>arXiv:191001499</i>.
    Published online 2019.
  apa: Ulirsch, M., &#38; Zakharov, D. (2019). Tropical double ramification loci.
    In <i>arXiv:1910.01499</i>.
  bibtex: '@article{Ulirsch_Zakharov_2019, title={Tropical double ramification loci},
    journal={arXiv:1910.01499}, author={Ulirsch, Martin and Zakharov, Dmitry}, year={2019}
    }'
  chicago: Ulirsch, Martin, and Dmitry Zakharov. “Tropical Double Ramification Loci.”
    <i>ArXiv:1910.01499</i>, 2019.
  ieee: M. Ulirsch and D. Zakharov, “Tropical double ramification loci,” <i>arXiv:1910.01499</i>.
    2019.
  mla: Ulirsch, Martin, and Dmitry Zakharov. “Tropical Double Ramification Loci.”
    <i>ArXiv:1910.01499</i>, 2019.
  short: M. Ulirsch, D. Zakharov, ArXiv:1910.01499 (2019).
date_created: 2026-07-08T06:49:12Z
date_updated: 2026-07-08T06:49:19Z
external_id:
  arxiv:
  - '1910.01499'
language:
- iso: eng
publication: arXiv:1910.01499
status: public
title: Tropical double ramification loci
type: preprint
user_id: '82981'
year: '2019'
...
---
_id: '66338'
author:
- first_name: Tyler
  full_name: Foster, Tyler
  last_name: Foster
- first_name: Dhruv
  full_name: Ranganathan, Dhruv
  last_name: Ranganathan
- first_name: Mattia
  full_name: Talpo, Mattia
  last_name: Talpo
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: Foster T, Ranganathan D, Talpo M, Ulirsch M. Logarithmic Picard groups, chip
    firing, and the combinatorial rank. <i>Mathematische Zeitschrift</i>. 2018;291(1-2):313-327.
    doi:<a href="https://doi.org/10.1007/s00209-018-2085-2">10.1007/s00209-018-2085-2</a>
  apa: Foster, T., Ranganathan, D., Talpo, M., &#38; Ulirsch, M. (2018). Logarithmic
    Picard groups, chip firing, and the combinatorial rank. <i>Mathematische Zeitschrift</i>,
    <i>291</i>(1–2), 313–327. <a href="https://doi.org/10.1007/s00209-018-2085-2">https://doi.org/10.1007/s00209-018-2085-2</a>
  bibtex: '@article{Foster_Ranganathan_Talpo_Ulirsch_2018, title={Logarithmic Picard
    groups, chip firing, and the combinatorial rank}, volume={291}, DOI={<a href="https://doi.org/10.1007/s00209-018-2085-2">10.1007/s00209-018-2085-2</a>},
    number={1–2}, journal={Mathematische Zeitschrift}, publisher={Springer Science
    and Business Media LLC}, author={Foster, Tyler and Ranganathan, Dhruv and Talpo,
    Mattia and Ulirsch, Martin}, year={2018}, pages={313–327} }'
  chicago: 'Foster, Tyler, Dhruv Ranganathan, Mattia Talpo, and Martin Ulirsch. “Logarithmic
    Picard Groups, Chip Firing, and the Combinatorial Rank.” <i>Mathematische Zeitschrift</i>
    291, no. 1–2 (2018): 313–27. <a href="https://doi.org/10.1007/s00209-018-2085-2">https://doi.org/10.1007/s00209-018-2085-2</a>.'
  ieee: 'T. Foster, D. Ranganathan, M. Talpo, and M. Ulirsch, “Logarithmic Picard
    groups, chip firing, and the combinatorial rank,” <i>Mathematische Zeitschrift</i>,
    vol. 291, no. 1–2, pp. 313–327, 2018, doi: <a href="https://doi.org/10.1007/s00209-018-2085-2">10.1007/s00209-018-2085-2</a>.'
  mla: Foster, Tyler, et al. “Logarithmic Picard Groups, Chip Firing, and the Combinatorial
    Rank.” <i>Mathematische Zeitschrift</i>, vol. 291, no. 1–2, Springer Science and
    Business Media LLC, 2018, pp. 313–27, doi:<a href="https://doi.org/10.1007/s00209-018-2085-2">10.1007/s00209-018-2085-2</a>.
  short: T. Foster, D. Ranganathan, M. Talpo, M. Ulirsch, Mathematische Zeitschrift
    291 (2018) 313–327.
date_created: 2026-07-08T07:08:39Z
date_updated: 2026-07-09T07:34:26Z
doi: 10.1007/s00209-018-2085-2
intvolume: '       291'
issue: 1-2
language:
- iso: eng
page: 313-327
publication: Mathematische Zeitschrift
publication_identifier:
  issn:
  - 0025-5874
  - 1432-1823
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Logarithmic Picard groups, chip firing, and the combinatorial rank
type: journal_article
user_id: '82981'
volume: 291
year: '2018'
...
---
_id: '66334'
author:
- first_name: Alana
  full_name: Huszar, Alana
  last_name: Huszar
- first_name: Steffen
  full_name: Marcus, Steffen
  last_name: Marcus
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: Huszar A, Marcus S, Ulirsch M. Clutching and gluing in tropical and logarithmic
    geometry. <i>Journal of Pure and Applied Algebra</i>. 2018;223(5):2036-2061. doi:<a
    href="https://doi.org/10.1016/j.jpaa.2018.08.014">10.1016/j.jpaa.2018.08.014</a>
  apa: Huszar, A., Marcus, S., &#38; Ulirsch, M. (2018). Clutching and gluing in tropical
    and logarithmic geometry. <i>Journal of Pure and Applied Algebra</i>, <i>223</i>(5),
    2036–2061. <a href="https://doi.org/10.1016/j.jpaa.2018.08.014">https://doi.org/10.1016/j.jpaa.2018.08.014</a>
  bibtex: '@article{Huszar_Marcus_Ulirsch_2018, title={Clutching and gluing in tropical
    and logarithmic geometry}, volume={223}, DOI={<a href="https://doi.org/10.1016/j.jpaa.2018.08.014">10.1016/j.jpaa.2018.08.014</a>},
    number={5}, journal={Journal of Pure and Applied Algebra}, publisher={Elsevier
    BV}, author={Huszar, Alana and Marcus, Steffen and Ulirsch, Martin}, year={2018},
    pages={2036–2061} }'
  chicago: 'Huszar, Alana, Steffen Marcus, and Martin Ulirsch. “Clutching and Gluing
    in Tropical and Logarithmic Geometry.” <i>Journal of Pure and Applied Algebra</i>
    223, no. 5 (2018): 2036–61. <a href="https://doi.org/10.1016/j.jpaa.2018.08.014">https://doi.org/10.1016/j.jpaa.2018.08.014</a>.'
  ieee: 'A. Huszar, S. Marcus, and M. Ulirsch, “Clutching and gluing in tropical and
    logarithmic geometry,” <i>Journal of Pure and Applied Algebra</i>, vol. 223, no.
    5, pp. 2036–2061, 2018, doi: <a href="https://doi.org/10.1016/j.jpaa.2018.08.014">10.1016/j.jpaa.2018.08.014</a>.'
  mla: Huszar, Alana, et al. “Clutching and Gluing in Tropical and Logarithmic Geometry.”
    <i>Journal of Pure and Applied Algebra</i>, vol. 223, no. 5, Elsevier BV, 2018,
    pp. 2036–61, doi:<a href="https://doi.org/10.1016/j.jpaa.2018.08.014">10.1016/j.jpaa.2018.08.014</a>.
  short: A. Huszar, S. Marcus, M. Ulirsch, Journal of Pure and Applied Algebra 223
    (2018) 2036–2061.
date_created: 2026-07-08T07:06:48Z
date_updated: 2026-07-09T06:50:37Z
doi: 10.1016/j.jpaa.2018.08.014
intvolume: '       223'
issue: '5'
language:
- iso: eng
page: 2036-2061
publication: Journal of Pure and Applied Algebra
publication_identifier:
  issn:
  - 0022-4049
publication_status: published
publisher: Elsevier BV
status: public
title: Clutching and gluing in tropical and logarithmic geometry
type: journal_article
user_id: '82981'
volume: 223
year: '2018'
...
---
_id: '66354'
author:
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: 'Ulirsch M. Functorial tropicalization of logarithmic schemes: the case of
    constant coefficients. <i>Proceedings of the London Mathematical Society</i>.
    2017;114(6):1081-1113. doi:<a href="https://doi.org/10.1112/plms.12031">10.1112/plms.12031</a>'
  apa: 'Ulirsch, M. (2017). Functorial tropicalization of logarithmic schemes: the
    case of constant coefficients. <i>Proceedings of the London Mathematical Society</i>,
    <i>114</i>(6), 1081–1113. <a href="https://doi.org/10.1112/plms.12031">https://doi.org/10.1112/plms.12031</a>'
  bibtex: '@article{Ulirsch_2017, title={Functorial tropicalization of logarithmic
    schemes: the case of constant coefficients}, volume={114}, DOI={<a href="https://doi.org/10.1112/plms.12031">10.1112/plms.12031</a>},
    number={6}, journal={Proceedings of the London Mathematical Society}, publisher={Wiley},
    author={Ulirsch, Martin}, year={2017}, pages={1081–1113} }'
  chicago: 'Ulirsch, Martin. “Functorial Tropicalization of Logarithmic Schemes: The
    Case of Constant Coefficients.” <i>Proceedings of the London Mathematical Society</i>
    114, no. 6 (2017): 1081–1113. <a href="https://doi.org/10.1112/plms.12031">https://doi.org/10.1112/plms.12031</a>.'
  ieee: 'M. Ulirsch, “Functorial tropicalization of logarithmic schemes: the case
    of constant coefficients,” <i>Proceedings of the London Mathematical Society</i>,
    vol. 114, no. 6, pp. 1081–1113, 2017, doi: <a href="https://doi.org/10.1112/plms.12031">10.1112/plms.12031</a>.'
  mla: 'Ulirsch, Martin. “Functorial Tropicalization of Logarithmic Schemes: The Case
    of Constant Coefficients.” <i>Proceedings of the London Mathematical Society</i>,
    vol. 114, no. 6, Wiley, 2017, pp. 1081–113, doi:<a href="https://doi.org/10.1112/plms.12031">10.1112/plms.12031</a>.'
  short: M. Ulirsch, Proceedings of the London Mathematical Society 114 (2017) 1081–1113.
date_created: 2026-07-08T07:24:50Z
date_updated: 2026-07-09T07:31:48Z
doi: 10.1112/plms.12031
intvolume: '       114'
issue: '6'
language:
- iso: eng
page: 1081-1113
publication: Proceedings of the London Mathematical Society
publication_identifier:
  issn:
  - 0024-6115
  - 1460-244X
publication_status: published
publisher: Wiley
status: public
title: 'Functorial tropicalization of logarithmic schemes: the case of constant coefficients'
type: journal_article
user_id: '82981'
volume: 114
year: '2017'
...
---
_id: '66337'
author:
- first_name: Bo
  full_name: Lin, Bo
  last_name: Lin
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: 'Lin B, Ulirsch M. Towards a tropical Hodge bundle. In: <i>Fields Institute
    Communications</i>. Springer New York; 2017. doi:<a href="https://doi.org/10.1007/978-1-4939-7486-3_16">10.1007/978-1-4939-7486-3_16</a>'
  apa: Lin, B., &#38; Ulirsch, M. (2017). Towards a tropical Hodge bundle. In <i>Fields
    Institute Communications</i>. Springer New York. <a href="https://doi.org/10.1007/978-1-4939-7486-3_16">https://doi.org/10.1007/978-1-4939-7486-3_16</a>
  bibtex: '@inbook{Lin_Ulirsch_2017, place={New York, NY}, title={Towards a tropical
    Hodge bundle}, DOI={<a href="https://doi.org/10.1007/978-1-4939-7486-3_16">10.1007/978-1-4939-7486-3_16</a>},
    booktitle={Fields Institute Communications}, publisher={Springer New York}, author={Lin,
    Bo and Ulirsch, Martin}, year={2017} }'
  chicago: 'Lin, Bo, and Martin Ulirsch. “Towards a Tropical Hodge Bundle.” In <i>Fields
    Institute Communications</i>. New York, NY: Springer New York, 2017. <a href="https://doi.org/10.1007/978-1-4939-7486-3_16">https://doi.org/10.1007/978-1-4939-7486-3_16</a>.'
  ieee: 'B. Lin and M. Ulirsch, “Towards a tropical Hodge bundle,” in <i>Fields Institute
    Communications</i>, New York, NY: Springer New York, 2017.'
  mla: Lin, Bo, and Martin Ulirsch. “Towards a Tropical Hodge Bundle.” <i>Fields Institute
    Communications</i>, Springer New York, 2017, doi:<a href="https://doi.org/10.1007/978-1-4939-7486-3_16">10.1007/978-1-4939-7486-3_16</a>.
  short: 'B. Lin, M. Ulirsch, in: Fields Institute Communications, Springer New York,
    New York, NY, 2017.'
date_created: 2026-07-08T07:08:13Z
date_updated: 2026-07-09T07:03:43Z
doi: 10.1007/978-1-4939-7486-3_16
language:
- iso: eng
place: New York, NY
publication: Fields Institute Communications
publication_identifier:
  isbn:
  - '9781493974856'
  - '9781493974863'
  issn:
  - 1069-5265
  - 2194-1564
publication_status: published
publisher: Springer New York
status: public
title: Towards a tropical Hodge bundle
type: book_chapter
user_id: '82981'
year: '2017'
...
---
_id: '66349'
author:
- first_name: Dan
  full_name: Abramovich, Dan
  last_name: Abramovich
- first_name: Qile
  full_name: Chen, Qile
  last_name: Chen
- first_name: Steffen
  full_name: Marcus, Steffen
  last_name: Marcus
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
- first_name: Jonathan
  full_name: Wise, Jonathan
  last_name: Wise
citation:
  ama: 'Abramovich D, Chen Q, Marcus S, Ulirsch M, Wise J. Skeletons and fans of logarithmic
    structures. In: <i>Simons Symposia</i>. Springer International Publishing; 2016.
    doi:<a href="https://doi.org/10.1007/978-3-319-30945-3_9">10.1007/978-3-319-30945-3_9</a>'
  apa: Abramovich, D., Chen, Q., Marcus, S., Ulirsch, M., &#38; Wise, J. (2016). Skeletons
    and fans of logarithmic structures. In <i>Simons Symposia</i>. Springer International
    Publishing. <a href="https://doi.org/10.1007/978-3-319-30945-3_9">https://doi.org/10.1007/978-3-319-30945-3_9</a>
  bibtex: '@inbook{Abramovich_Chen_Marcus_Ulirsch_Wise_2016, place={Cham}, title={Skeletons
    and fans of logarithmic structures}, DOI={<a href="https://doi.org/10.1007/978-3-319-30945-3_9">10.1007/978-3-319-30945-3_9</a>},
    booktitle={Simons Symposia}, publisher={Springer International Publishing}, author={Abramovich,
    Dan and Chen, Qile and Marcus, Steffen and Ulirsch, Martin and Wise, Jonathan},
    year={2016} }'
  chicago: 'Abramovich, Dan, Qile Chen, Steffen Marcus, Martin Ulirsch, and Jonathan
    Wise. “Skeletons and Fans of Logarithmic Structures.” In <i>Simons Symposia</i>.
    Cham: Springer International Publishing, 2016. <a href="https://doi.org/10.1007/978-3-319-30945-3_9">https://doi.org/10.1007/978-3-319-30945-3_9</a>.'
  ieee: 'D. Abramovich, Q. Chen, S. Marcus, M. Ulirsch, and J. Wise, “Skeletons and
    fans of logarithmic structures,” in <i>Simons Symposia</i>, Cham: Springer International
    Publishing, 2016.'
  mla: Abramovich, Dan, et al. “Skeletons and Fans of Logarithmic Structures.” <i>Simons
    Symposia</i>, Springer International Publishing, 2016, doi:<a href="https://doi.org/10.1007/978-3-319-30945-3_9">10.1007/978-3-319-30945-3_9</a>.
  short: 'D. Abramovich, Q. Chen, S. Marcus, M. Ulirsch, J. Wise, in: Simons Symposia,
    Springer International Publishing, Cham, 2016.'
date_created: 2026-07-08T07:22:18Z
date_updated: 2026-07-09T06:52:38Z
doi: 10.1007/978-3-319-30945-3_9
language:
- iso: eng
place: Cham
publication: Simons Symposia
publication_identifier:
  isbn:
  - '9783319309446'
  - '9783319309453'
  issn:
  - 2365-9564
  - 2365-9572
publication_status: published
publisher: Springer International Publishing
status: public
title: Skeletons and fans of logarithmic structures
type: book_chapter
user_id: '82981'
year: '2016'
...
---
_id: '66362'
author:
- first_name: Man-Wai
  full_name: Cheung, Man-Wai
  last_name: Cheung
- first_name: Lorenzo
  full_name: Fantini, Lorenzo
  last_name: Fantini
- first_name: Jennifer
  full_name: Park, Jennifer
  last_name: Park
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: Cheung M-W, Fantini L, Park J, Ulirsch M. Faithful Realizability of Tropical
    Curves. <i>International Mathematics Research Notices</i>. 2015;2016(15):4706-4727.
    doi:<a href="https://doi.org/10.1093/imrn/rnv269">10.1093/imrn/rnv269</a>
  apa: Cheung, M.-W., Fantini, L., Park, J., &#38; Ulirsch, M. (2015). Faithful Realizability
    of Tropical Curves. <i>International Mathematics Research Notices</i>, <i>2016</i>(15),
    4706–4727. <a href="https://doi.org/10.1093/imrn/rnv269">https://doi.org/10.1093/imrn/rnv269</a>
  bibtex: '@article{Cheung_Fantini_Park_Ulirsch_2015, title={Faithful Realizability
    of Tropical Curves}, volume={2016}, DOI={<a href="https://doi.org/10.1093/imrn/rnv269">10.1093/imrn/rnv269</a>},
    number={15}, journal={International Mathematics Research Notices}, publisher={Oxford
    University Press (OUP)}, author={Cheung, Man-Wai and Fantini, Lorenzo and Park,
    Jennifer and Ulirsch, Martin}, year={2015}, pages={4706–4727} }'
  chicago: 'Cheung, Man-Wai, Lorenzo Fantini, Jennifer Park, and Martin Ulirsch. “Faithful
    Realizability of Tropical Curves.” <i>International Mathematics Research Notices</i>
    2016, no. 15 (2015): 4706–27. <a href="https://doi.org/10.1093/imrn/rnv269">https://doi.org/10.1093/imrn/rnv269</a>.'
  ieee: 'M.-W. Cheung, L. Fantini, J. Park, and M. Ulirsch, “Faithful Realizability
    of Tropical Curves,” <i>International Mathematics Research Notices</i>, vol. 2016,
    no. 15, pp. 4706–4727, 2015, doi: <a href="https://doi.org/10.1093/imrn/rnv269">10.1093/imrn/rnv269</a>.'
  mla: Cheung, Man-Wai, et al. “Faithful Realizability of Tropical Curves.” <i>International
    Mathematics Research Notices</i>, vol. 2016, no. 15, Oxford University Press (OUP),
    2015, pp. 4706–27, doi:<a href="https://doi.org/10.1093/imrn/rnv269">10.1093/imrn/rnv269</a>.
  short: M.-W. Cheung, L. Fantini, J. Park, M. Ulirsch, International Mathematics
    Research Notices 2016 (2015) 4706–4727.
date_created: 2026-07-08T08:38:51Z
date_updated: 2026-07-08T08:39:05Z
doi: 10.1093/imrn/rnv269
intvolume: '      2016'
issue: '15'
language:
- iso: eng
page: 4706-4727
publication: International Mathematics Research Notices
publication_identifier:
  issn:
  - 1073-7928
  - 1687-0247
publication_status: published
publisher: Oxford University Press (OUP)
status: public
title: Faithful Realizability of Tropical Curves
type: journal_article
user_id: '82981'
volume: 2016
year: '2015'
...
---
_id: '66355'
author:
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: Ulirsch M. Tropical compactification in log-regular varieties. <i>Mathematische
    Zeitschrift</i>. 2015;280(1-2):195-210. doi:<a href="https://doi.org/10.1007/s00209-015-1418-7">10.1007/s00209-015-1418-7</a>
  apa: Ulirsch, M. (2015). Tropical compactification in log-regular varieties. <i>Mathematische
    Zeitschrift</i>, <i>280</i>(1–2), 195–210. <a href="https://doi.org/10.1007/s00209-015-1418-7">https://doi.org/10.1007/s00209-015-1418-7</a>
  bibtex: '@article{Ulirsch_2015, title={Tropical compactification in log-regular
    varieties}, volume={280}, DOI={<a href="https://doi.org/10.1007/s00209-015-1418-7">10.1007/s00209-015-1418-7</a>},
    number={1–2}, journal={Mathematische Zeitschrift}, publisher={Springer Science
    and Business Media LLC}, author={Ulirsch, Martin}, year={2015}, pages={195–210}
    }'
  chicago: 'Ulirsch, Martin. “Tropical Compactification in Log-Regular Varieties.”
    <i>Mathematische Zeitschrift</i> 280, no. 1–2 (2015): 195–210. <a href="https://doi.org/10.1007/s00209-015-1418-7">https://doi.org/10.1007/s00209-015-1418-7</a>.'
  ieee: 'M. Ulirsch, “Tropical compactification in log-regular varieties,” <i>Mathematische
    Zeitschrift</i>, vol. 280, no. 1–2, pp. 195–210, 2015, doi: <a href="https://doi.org/10.1007/s00209-015-1418-7">10.1007/s00209-015-1418-7</a>.'
  mla: Ulirsch, Martin. “Tropical Compactification in Log-Regular Varieties.” <i>Mathematische
    Zeitschrift</i>, vol. 280, no. 1–2, Springer Science and Business Media LLC, 2015,
    pp. 195–210, doi:<a href="https://doi.org/10.1007/s00209-015-1418-7">10.1007/s00209-015-1418-7</a>.
  short: M. Ulirsch, Mathematische Zeitschrift 280 (2015) 195–210.
date_created: 2026-07-08T07:25:16Z
date_updated: 2026-07-09T07:00:13Z
doi: 10.1007/s00209-015-1418-7
intvolume: '       280'
issue: 1-2
language:
- iso: eng
page: 195-210
publication: Mathematische Zeitschrift
publication_identifier:
  issn:
  - 0025-5874
  - 1432-1823
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Tropical compactification in log-regular varieties
type: journal_article
user_id: '82981'
volume: 280
year: '2015'
...
---
_id: '66352'
author:
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: Ulirsch M. Tropical geometry of moduli spaces of weighted stable curves. <i>Journal
    of the London Mathematical Society</i>. 2015;92(2):427-450. doi:<a href="https://doi.org/10.1112/jlms/jdv032">10.1112/jlms/jdv032</a>
  apa: Ulirsch, M. (2015). Tropical geometry of moduli spaces of weighted stable curves.
    <i>Journal of the London Mathematical Society</i>, <i>92</i>(2), 427–450. <a href="https://doi.org/10.1112/jlms/jdv032">https://doi.org/10.1112/jlms/jdv032</a>
  bibtex: '@article{Ulirsch_2015, title={Tropical geometry of moduli spaces of weighted
    stable curves}, volume={92}, DOI={<a href="https://doi.org/10.1112/jlms/jdv032">10.1112/jlms/jdv032</a>},
    number={2}, journal={Journal of the London Mathematical Society}, publisher={Wiley},
    author={Ulirsch, Martin}, year={2015}, pages={427–450} }'
  chicago: 'Ulirsch, Martin. “Tropical Geometry of Moduli Spaces of Weighted Stable
    Curves.” <i>Journal of the London Mathematical Society</i> 92, no. 2 (2015): 427–50.
    <a href="https://doi.org/10.1112/jlms/jdv032">https://doi.org/10.1112/jlms/jdv032</a>.'
  ieee: 'M. Ulirsch, “Tropical geometry of moduli spaces of weighted stable curves,”
    <i>Journal of the London Mathematical Society</i>, vol. 92, no. 2, pp. 427–450,
    2015, doi: <a href="https://doi.org/10.1112/jlms/jdv032">10.1112/jlms/jdv032</a>.'
  mla: Ulirsch, Martin. “Tropical Geometry of Moduli Spaces of Weighted Stable Curves.”
    <i>Journal of the London Mathematical Society</i>, vol. 92, no. 2, Wiley, 2015,
    pp. 427–50, doi:<a href="https://doi.org/10.1112/jlms/jdv032">10.1112/jlms/jdv032</a>.
  short: M. Ulirsch, Journal of the London Mathematical Society 92 (2015) 427–450.
date_created: 2026-07-08T07:24:26Z
date_updated: 2026-07-09T07:01:53Z
doi: 10.1112/jlms/jdv032
intvolume: '        92'
issue: '2'
language:
- iso: eng
page: 427-450
publication: Journal of the London Mathematical Society
publication_identifier:
  issn:
  - 0024-6107
  - 1469-7750
publication_status: published
publisher: Wiley
status: public
title: Tropical geometry of moduli spaces of weighted stable curves
type: journal_article
user_id: '82981'
volume: 92
year: '2015'
...