---
_id: '13091'
abstract:
- lang: eng
  text: We use a function field version of the Hardy–Littlewood circle method to study
    the locus of free rational curves on an arbitrary smooth projective hypersurface
    of sufficiently low degree. On the one hand this allows us to bound the dimension
    of the singular locus of the moduli space of rational curves on such hypersurfaces
    and, on the other hand, it sheds light on Peyre’s reformulation of the Batyrev–Manin
    conjecture in terms of slopes with respect to the tangent bundle.
acknowledgement: The authors are grateful to Paul Nelson, Per Salberger and Jason
  Starr for useful comments. While working on this paper the first author was supported
  by EPRSC grant EP/P026710/1. The research was partially conducted during the period
  the second author served as a Clay Research Fellow, and partially conducted during
  the period he was supported by Dr. Max Rössler, the Walter Haefner Foundation and
  the ETH Zurich Foundation.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
- first_name: Will
  full_name: Sawin, Will
  last_name: Sawin
citation:
  ama: Browning TD, Sawin W. Free rational curves on low degree hypersurfaces and
    the circle method. <i>Algebra and Number Theory</i>. 2023;17(3):719-748. doi:<a
    href="https://doi.org/10.2140/ant.2023.17.719">10.2140/ant.2023.17.719</a>
  apa: Browning, T. D., &#38; Sawin, W. (2023). Free rational curves on low degree
    hypersurfaces and the circle method. <i>Algebra and Number Theory</i>. Mathematical
    Sciences Publishers. <a href="https://doi.org/10.2140/ant.2023.17.719">https://doi.org/10.2140/ant.2023.17.719</a>
  chicago: Browning, Timothy D, and Will Sawin. “Free Rational Curves on Low Degree
    Hypersurfaces and the Circle Method.” <i>Algebra and Number Theory</i>. Mathematical
    Sciences Publishers, 2023. <a href="https://doi.org/10.2140/ant.2023.17.719">https://doi.org/10.2140/ant.2023.17.719</a>.
  ieee: T. D. Browning and W. Sawin, “Free rational curves on low degree hypersurfaces
    and the circle method,” <i>Algebra and Number Theory</i>, vol. 17, no. 3. Mathematical
    Sciences Publishers, pp. 719–748, 2023.
  ista: Browning TD, Sawin W. 2023. Free rational curves on low degree hypersurfaces
    and the circle method. Algebra and Number Theory. 17(3), 719–748.
  mla: Browning, Timothy D., and Will Sawin. “Free Rational Curves on Low Degree Hypersurfaces
    and the Circle Method.” <i>Algebra and Number Theory</i>, vol. 17, no. 3, Mathematical
    Sciences Publishers, 2023, pp. 719–48, doi:<a href="https://doi.org/10.2140/ant.2023.17.719">10.2140/ant.2023.17.719</a>.
  short: T.D. Browning, W. Sawin, Algebra and Number Theory 17 (2023) 719–748.
date_created: 2023-05-28T22:01:02Z
date_published: 2023-04-12T00:00:00Z
date_updated: 2023-08-01T14:51:57Z
day: '12'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.2140/ant.2023.17.719
external_id:
  arxiv:
  - '1810.06882'
  isi:
  - '000996014700004'
file:
- access_level: open_access
  checksum: 5d5d67b235905650e33cf7065d7583b4
  content_type: application/pdf
  creator: dernst
  date_created: 2023-05-30T08:05:22Z
  date_updated: 2023-05-30T08:05:22Z
  file_id: '13101'
  file_name: 2023_AlgebraNumberTheory_Browning.pdf
  file_size: 1430719
  relation: main_file
  success: 1
file_date_updated: 2023-05-30T08:05:22Z
has_accepted_license: '1'
intvolume: '        17'
isi: 1
issue: '3'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 719-748
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
  grant_number: EP-P026710-2
  name: Between rational and integral points
publication: Algebra and Number Theory
publication_identifier:
  eissn:
  - 1944-7833
  issn:
  - 1937-0652
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Free rational curves on low degree hypersurfaces and the circle method
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 17
year: '2023'
...
---
_id: '9199'
abstract:
- lang: eng
  text: "We associate a certain tensor product lattice to any primitive integer lattice
    and ask about its typical shape. These lattices are related to the tangent bundle
    of Grassmannians and their study is motivated by Peyre's programme on \"freeness\"
    for rational points of bounded height on Fano\r\nvarieties."
acknowledgement: The authors are very grateful to Will Sawin for useful remarks about
  this topic. While working on this paper the first two authors were supported by
  EPSRC grant EP/P026710/1, and the first and last authors by FWF grant P 32428-N35.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
- first_name: Tal
  full_name: Horesh, Tal
  id: C8B7BF48-8D81-11E9-BCA9-F536E6697425
  last_name: Horesh
- first_name: Florian Alexander
  full_name: Wilsch, Florian Alexander
  id: 560601DA-8D36-11E9-A136-7AC1E5697425
  last_name: Wilsch
  orcid: 0000-0001-7302-8256
citation:
  ama: Browning TD, Horesh T, Wilsch FA. Equidistribution and freeness on Grassmannians.
    <i>Algebra &#38; Number Theory</i>. 2022;16(10):2385-2407. doi:<a href="https://doi.org/10.2140/ant.2022.16.2385">10.2140/ant.2022.16.2385</a>
  apa: Browning, T. D., Horesh, T., &#38; Wilsch, F. A. (2022). Equidistribution and
    freeness on Grassmannians. <i>Algebra &#38; Number Theory</i>. Mathematical Sciences
    Publishers. <a href="https://doi.org/10.2140/ant.2022.16.2385">https://doi.org/10.2140/ant.2022.16.2385</a>
  chicago: Browning, Timothy D, Tal Horesh, and Florian Alexander Wilsch. “Equidistribution
    and Freeness on Grassmannians.” <i>Algebra &#38; Number Theory</i>. Mathematical
    Sciences Publishers, 2022. <a href="https://doi.org/10.2140/ant.2022.16.2385">https://doi.org/10.2140/ant.2022.16.2385</a>.
  ieee: T. D. Browning, T. Horesh, and F. A. Wilsch, “Equidistribution and freeness
    on Grassmannians,” <i>Algebra &#38; Number Theory</i>, vol. 16, no. 10. Mathematical
    Sciences Publishers, pp. 2385–2407, 2022.
  ista: Browning TD, Horesh T, Wilsch FA. 2022. Equidistribution and freeness on Grassmannians.
    Algebra &#38; Number Theory. 16(10), 2385–2407.
  mla: Browning, Timothy D., et al. “Equidistribution and Freeness on Grassmannians.”
    <i>Algebra &#38; Number Theory</i>, vol. 16, no. 10, Mathematical Sciences Publishers,
    2022, pp. 2385–407, doi:<a href="https://doi.org/10.2140/ant.2022.16.2385">10.2140/ant.2022.16.2385</a>.
  short: T.D. Browning, T. Horesh, F.A. Wilsch, Algebra &#38; Number Theory 16 (2022)
    2385–2407.
date_created: 2021-02-25T09:56:57Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2023-08-02T06:46:38Z
day: '01'
department:
- _id: TiBr
doi: 10.2140/ant.2022.16.2385
external_id:
  arxiv:
  - '2102.11552'
  isi:
  - '000961514100004'
intvolume: '        16'
isi: 1
issue: '10'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2102.11552
month: '12'
oa: 1
oa_version: Preprint
page: 2385-2407
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
  grant_number: EP-P026710-2
  name: Between rational and integral points
- _id: 26AEDAB2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P32428
  name: New frontiers of the Manin conjecture
publication: Algebra & Number Theory
publication_identifier:
  eissn:
  - 1944-7833
  issn:
  - 1937-0652
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Equidistribution and freeness on Grassmannians
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 16
year: '2022'
...