---
OA_place: publisher
OA_type: diamond
PlanS_conform: '1'
_id: '19054'
abstract:
- lang: eng
  text: "This work concerns asymptotical stabilisation phenomena occurring in the
    moduli space of sections of certain algebraic families over a smooth projective
    curve, whenever the generic fibre of the family is a smooth projective Fano variety,
    or not far from being Fano.\r\n We describe the expected behaviour of the class,
    in a ring of motivic integration, of the moduli space of sections of given numerical
    class. Up to an adequate normalisation, it should converge, when the class of
    the sections goes arbitrarily far from the boundary of the dual of the effective
    cone, to an effective element given by a motivic Euler product. Such a principle
    can be seen as an analogue for rational curves of the Batyrev-Manin-Peyre principle
    for rational points.\r\n The central tool of this article is the property of equidistribution
    of curves. We show that this notion does not depend on the choice of a model of
    the generic fibre, and that equidistribution of curves holds for smooth projective
    split toric varieties. As an application, we study the Batyrev-Manin-Peyre principle
    for curves on a certain kind of twisted products."
acknowledgement: I am very grateful to my Ph.D. advisor Emmanuel Peyre for all the
  remarks and suggestions he made during the writing of this article. I warmly thank
  Margaret Bilu and Tim Browning for some valuable comments they made on a preliminary
  version of this work. I would like to thank David Bourqui as well for several helpful
  conversations. Finally, I thank the anonymous referee for their very careful reading
  and their numerous comments and suggestions which helped me a lot in improving the
  exposition, besides fixing several typos, and Elizabeth Weaver for the final editing
  work. During the revision process of this work, the author received funding from
  the European Union’s Horizon 2020 research and innovation programme under the Marie
  Skłodowska-Curie Grant Agreement No. 101034413.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Loïs
  full_name: Faisant, Loïs
  id: 26ca6926-5797-11ee-9232-f8b51bd19631
  last_name: Faisant
citation:
  ama: Faisant L. Motivic distribution of rational curves and twisted products of
    toric varieties. <i>Algebra &#38; Number Theory</i>. 2025;19:883-965. doi:<a href="https://doi.org/10.2140/ant.2025.19.883">10.2140/ant.2025.19.883</a>
  apa: Faisant, L. (2025). Motivic distribution of rational curves and twisted products
    of toric varieties. <i>Algebra &#38; Number Theory</i>. Mathematical Sciences
    Publishers. <a href="https://doi.org/10.2140/ant.2025.19.883">https://doi.org/10.2140/ant.2025.19.883</a>
  chicago: Faisant, Loïs. “Motivic Distribution of Rational Curves and Twisted Products
    of Toric Varieties.” <i>Algebra &#38; Number Theory</i>. Mathematical Sciences
    Publishers, 2025. <a href="https://doi.org/10.2140/ant.2025.19.883">https://doi.org/10.2140/ant.2025.19.883</a>.
  ieee: L. Faisant, “Motivic distribution of rational curves and twisted products
    of toric varieties,” <i>Algebra &#38; Number Theory</i>, vol. 19. Mathematical
    Sciences Publishers, pp. 883–965, 2025.
  ista: Faisant L. 2025. Motivic distribution of rational curves and twisted products
    of toric varieties. Algebra &#38; Number Theory. 19, 883–965.
  mla: Faisant, Loïs. “Motivic Distribution of Rational Curves and Twisted Products
    of Toric Varieties.” <i>Algebra &#38; Number Theory</i>, vol. 19, Mathematical
    Sciences Publishers, 2025, pp. 883–965, doi:<a href="https://doi.org/10.2140/ant.2025.19.883">10.2140/ant.2025.19.883</a>.
  short: L. Faisant, Algebra &#38; Number Theory 19 (2025) 883–965.
corr_author: '1'
date_created: 2025-02-18T13:33:14Z
date_published: 2025-04-22T00:00:00Z
date_updated: 2026-02-17T13:19:19Z
day: '22'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.2140/ant.2025.19.883
ec_funded: 1
external_id:
  arxiv:
  - '2302.07339'
file:
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license: https://creativecommons.org/licenses/by/4.0/
month: '04'
oa: 1
oa_version: Published Version
page: 883-965
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
  call_identifier: H2020
  grant_number: '101034413'
  name: 'IST-BRIDGE: International postdoctoral program'
publication: Algebra & Number Theory
publication_identifier:
  eissn:
  - 1944-7833
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
status: public
title: Motivic distribution of rational curves and twisted products of toric varieties
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type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 19
year: '2025'
...
---
OA_place: publisher
OA_type: diamond
PlanS_conform: '1'
_id: '21244'
abstract:
- lang: eng
  text: 'Given a family of varieties over the projective line, we study the density
    of fibres that are everywhere locally soluble in the case that components of higher
    multiplicity are allowed. We use log geometry to formulate a new sparsity criterion
    for the existence of everywhere locally soluble fibres and formulate new conjectures
    that generalise previous work of Loughran and Smeets. These conjectures involve
    geometric invariants of the associated multiplicity orbifolds on the base of the
    fibration in the spirit of Campana. We give evidence for the conjectures by providing
    an assortment of bounds using Chebotarev’s theorem and sieve methods, with most
    of the evidence involving upper bounds. '
acknowledgement: We are very grateful to Tim Santens for useful conversations and
  to the anonymous referees for numerous pertinent remarks. While working on this
  paper, Browning was supported by a FWF grant (DOI 10.55776/P32428), Lyczak was supported
  by UKRI MR/V021362/1, and Smeets was supported by grant G0B1721N of the Fund for
  Scientific Research – Flanders.
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: Julian
  full_name: Lyczak, Julian
  last_name: Lyczak
- first_name: Arne
  full_name: Smeets, Arne
  last_name: Smeets
citation:
  ama: Browning TD, Lyczak J, Smeets A. Paucity of rational points on fibrations with
    multiple fibres. <i>Algebra &#38; Number Theory</i>. 2025;19(10):2049-2090. doi:<a
    href="https://doi.org/10.2140/ant.2025.19.2049">10.2140/ant.2025.19.2049</a>
  apa: Browning, T. D., Lyczak, J., &#38; Smeets, A. (2025). Paucity of rational points
    on fibrations with multiple fibres. <i>Algebra &#38; Number Theory</i>. Mathematical
    Sciences Publishers. <a href="https://doi.org/10.2140/ant.2025.19.2049">https://doi.org/10.2140/ant.2025.19.2049</a>
  chicago: Browning, Timothy D, Julian Lyczak, and Arne Smeets. “Paucity of Rational
    Points on Fibrations with Multiple Fibres.” <i>Algebra &#38; Number Theory</i>.
    Mathematical Sciences Publishers, 2025. <a href="https://doi.org/10.2140/ant.2025.19.2049">https://doi.org/10.2140/ant.2025.19.2049</a>.
  ieee: T. D. Browning, J. Lyczak, and A. Smeets, “Paucity of rational points on fibrations
    with multiple fibres,” <i>Algebra &#38; Number Theory</i>, vol. 19, no. 10. Mathematical
    Sciences Publishers, pp. 2049–2090, 2025.
  ista: Browning TD, Lyczak J, Smeets A. 2025. Paucity of rational points on fibrations
    with multiple fibres. Algebra &#38; Number Theory. 19(10), 2049–2090.
  mla: Browning, Timothy D., et al. “Paucity of Rational Points on Fibrations with
    Multiple Fibres.” <i>Algebra &#38; Number Theory</i>, vol. 19, no. 10, Mathematical
    Sciences Publishers, 2025, pp. 2049–90, doi:<a href="https://doi.org/10.2140/ant.2025.19.2049">10.2140/ant.2025.19.2049</a>.
  short: T.D. Browning, J. Lyczak, A. Smeets, Algebra &#38; Number Theory 19 (2025)
    2049–2090.
corr_author: '1'
date_created: 2026-02-16T15:22:19Z
date_published: 2025-09-05T00:00:00Z
date_updated: 2026-02-17T11:59:57Z
day: '05'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.2140/ant.2025.19.2049
external_id:
  arxiv:
  - '2310.01135'
file:
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  checksum: e50a60a4303b81563f7adbcadbe2e986
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  date_created: 2026-02-17T11:56:20Z
  date_updated: 2026-02-17T11:56:20Z
  file_id: '21300'
  file_name: 2025_AlgebraNumberTheory_Browning.pdf
  file_size: 1505580
  relation: main_file
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has_accepted_license: '1'
intvolume: '        19'
issue: '10'
language:
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month: '09'
oa: 1
oa_version: Published Version
page: 2049-2090
project:
- _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'
status: public
title: Paucity of rational points on fibrations with multiple fibres
tmp:
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  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 19
year: '2025'
...
---
_id: '17449'
abstract:
- lang: eng
  text: We prove that the $k$-th positive integer moment of partial sums of Steinhaus
    random multiplicative functions over the interval $(x, x+H]$ matches the corresponding
    Gaussian moment, as long as $H\ll x/(\log x)^{2k^2+2+o(1)}$ and $H$ tends to infinity
    with $x$. We show that properly normalized partial sums of typical multiplicative
    functions arising from realizations of random multiplicative functions have Gaussian
    limiting distribution in short moving intervals $(x, x+H]$ with $H\ll X/(\log
    X)^{W(X)}$ tending to infinity with $X$, where $x$ is uniformly chosen from $\{1,2,\dots,
    X\}$, and $W(X)$ tends to infinity with $X$ arbitrarily slowly. This makes some
    initial progress on a recent question of Harper.
acknowledgement: "We thank Andrew Granville and the anonymous referee for many detailed
  comments that led us to significantly improve the results and presentation of our
  work. We thank and Adam Harper for helpful discussions and useful comments and corrections
  on earlier versions. We also thank Yuqiu Fu, Larry Guth, Kannan Soundararajan, Katharine
  Woo, and Liyang Yang for helpful discussions. Finally, we thank Peter Sarnak for
  introducing us (the authors) to each other during the “50 Years of Number Theory
  and Random Matrix Theory” Conference at IAS and making the collaboration possible.
  \r\nOpen Access made possible by participating institutions via Subscribe to Open."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Mayank
  full_name: Pandey, Mayank
  last_name: Pandey
- first_name: Victor
  full_name: Wang, Victor
  id: 76096395-aea4-11ed-a680-ab8ebbd3f1b9
  last_name: Wang
  orcid: 0000-0002-0704-7026
- first_name: Max Wenqiang
  full_name: Xu, Max Wenqiang
  last_name: Xu
citation:
  ama: Pandey M, Wang V, Xu MW. Partial sums of typical multiplicative functions over
    short moving intervals. <i>Algebra &#38; Number Theory</i>. 2024;18(2):389-408.
    doi:<a href="https://doi.org/10.2140/ant.2024.18.389">10.2140/ant.2024.18.389</a>
  apa: Pandey, M., Wang, V., &#38; Xu, M. W. (2024). Partial sums of typical multiplicative
    functions over short moving intervals. <i>Algebra &#38; Number Theory</i>. Mathematical
    Sciences Publishers. <a href="https://doi.org/10.2140/ant.2024.18.389">https://doi.org/10.2140/ant.2024.18.389</a>
  chicago: Pandey, Mayank, Victor Wang, and Max Wenqiang Xu. “Partial Sums of Typical
    Multiplicative Functions over Short Moving Intervals.” <i>Algebra &#38; Number
    Theory</i>. Mathematical Sciences Publishers, 2024. <a href="https://doi.org/10.2140/ant.2024.18.389">https://doi.org/10.2140/ant.2024.18.389</a>.
  ieee: M. Pandey, V. Wang, and M. W. Xu, “Partial sums of typical multiplicative
    functions over short moving intervals,” <i>Algebra &#38; Number Theory</i>, vol.
    18, no. 2. Mathematical Sciences Publishers, pp. 389–408, 2024.
  ista: Pandey M, Wang V, Xu MW. 2024. Partial sums of typical multiplicative functions
    over short moving intervals. Algebra &#38; Number Theory. 18(2), 389–408.
  mla: Pandey, Mayank, et al. “Partial Sums of Typical Multiplicative Functions over
    Short Moving Intervals.” <i>Algebra &#38; Number Theory</i>, vol. 18, no. 2, Mathematical
    Sciences Publishers, 2024, pp. 389–408, doi:<a href="https://doi.org/10.2140/ant.2024.18.389">10.2140/ant.2024.18.389</a>.
  short: M. Pandey, V. Wang, M.W. Xu, Algebra &#38; Number Theory 18 (2024) 389–408.
date_created: 2024-08-20T08:48:26Z
date_published: 2024-02-06T00:00:00Z
date_updated: 2024-08-21T06:58:43Z
day: '06'
ddc:
- '510'
doi: 10.2140/ant.2024.18.389
extern: '1'
external_id:
  arxiv:
  - '2207.11758'
file:
- access_level: open_access
  checksum: 1e3467a14de754bf8d3bff03a015e1ce
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  date_created: 2024-08-21T06:46:56Z
  date_updated: 2024-08-21T06:46:56Z
  file_id: '17455'
  file_name: 2024_AlgebraNumberTheory_Pandey.pdf
  file_size: 1401725
  relation: main_file
  success: 1
file_date_updated: 2024-08-21T06:46:56Z
has_accepted_license: '1'
intvolume: '        18'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2207.11758
month: '02'
oa: 1
oa_version: Published Version
page: 389-408
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: Partial sums of typical multiplicative functions over short moving intervals
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  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 18
year: '2024'
...
---
_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.
corr_author: '1'
date_created: 2023-05-28T22:01:02Z
date_published: 2023-04-12T00:00:00Z
date_updated: 2025-04-14T09:25:44Z
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.
corr_author: '1'
date_created: 2021-02-25T09:56:57Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2025-04-14T09:25:44Z
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'
...
---
_id: '15279'
abstract:
- lang: eng
  text: We formulate and prove an analog of Poonen’s finite-field Bertini theorem
    with Taylor conditions that holds in the Grothendieck ring of varieties. This
    gives a broad generalization of the work of Vakil and Wood, who treated the case
    of smooth hypersurface sections, and is made possible by the use of motivic Euler
    products to write down candidate motivic probabilities. As applications, we give
    motivic analogs of many results in arithmetic statistics that have been proven
    using Poonen’s sieve, including work of Bucur and Kedlaya on complete intersections
    and Erman and Wood on semiample Bertini theorems.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Margaret
  full_name: Bilu, Margaret
  id: 98C47862-10D5-11EA-BEDD-0F6F3DDC885E
  last_name: Bilu
- first_name: Sean
  full_name: Howe, Sean
  last_name: Howe
citation:
  ama: Bilu M, Howe S. Motivic Euler products in motivic statistics. <i>Algebra &#38;
    Number Theory</i>. 2021;15(9):2195-2259. doi:<a href="https://doi.org/10.2140/ant.2021.15.2195">10.2140/ant.2021.15.2195</a>
  apa: Bilu, M., &#38; Howe, S. (2021). Motivic Euler products in motivic statistics.
    <i>Algebra &#38; Number Theory</i>. Mathematical Sciences Publishers. <a href="https://doi.org/10.2140/ant.2021.15.2195">https://doi.org/10.2140/ant.2021.15.2195</a>
  chicago: Bilu, Margaret, and Sean Howe. “Motivic Euler Products in Motivic Statistics.”
    <i>Algebra &#38; Number Theory</i>. Mathematical Sciences Publishers, 2021. <a
    href="https://doi.org/10.2140/ant.2021.15.2195">https://doi.org/10.2140/ant.2021.15.2195</a>.
  ieee: M. Bilu and S. Howe, “Motivic Euler products in motivic statistics,” <i>Algebra
    &#38; Number Theory</i>, vol. 15, no. 9. Mathematical Sciences Publishers, pp.
    2195–2259, 2021.
  ista: Bilu M, Howe S. 2021. Motivic Euler products in motivic statistics. Algebra
    &#38; Number Theory. 15(9), 2195–2259.
  mla: Bilu, Margaret, and Sean Howe. “Motivic Euler Products in Motivic Statistics.”
    <i>Algebra &#38; Number Theory</i>, vol. 15, no. 9, Mathematical Sciences Publishers,
    2021, pp. 2195–259, doi:<a href="https://doi.org/10.2140/ant.2021.15.2195">10.2140/ant.2021.15.2195</a>.
  short: M. Bilu, S. Howe, Algebra &#38; Number Theory 15 (2021) 2195–2259.
corr_author: '1'
date_created: 2024-04-03T08:12:59Z
date_published: 2021-12-23T00:00:00Z
date_updated: 2024-10-21T06:02:15Z
day: '23'
department:
- _id: TiBr
doi: 10.2140/ant.2021.15.2195
external_id:
  arxiv:
  - '1910.05207'
intvolume: '        15'
issue: '9'
keyword:
- Algebra and Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1910.05207
month: '12'
oa: 1
oa_version: Preprint
page: 2195-2259
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: Motivic Euler products in motivic statistics
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 15
year: '2021'
...
---
_id: '265'
abstract:
- lang: eng
  text: We establish the dimension and irreducibility of the moduli space of rational
    curves (of fixed degree) on arbitrary smooth hypersurfaces of sufficiently low
    degree. A spreading out argument reduces the problem to hypersurfaces defined
    over finite fields of large cardinality, which can then be tackled using a function
    field version of the Hardy-Littlewood circle method, in which particular care
    is taken to ensure uniformity in the size of the underlying finite field.
acknowledgement: While working on this paper the first author was supported by ERC
  grant 306457.
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: Pankaj
  full_name: Vishe, Pankaj
  last_name: Vishe
citation:
  ama: Browning TD, Vishe P. Rational curves on smooth hypersurfaces of low degree.
    <i>Geometric Methods in Algebra and Number Theory</i>. 2017;11(7):1657-1675. doi:<a
    href="https://doi.org/10.2140/ant.2017.11.1657">10.2140/ant.2017.11.1657</a>
  apa: Browning, T. D., &#38; Vishe, P. (2017). Rational curves on smooth hypersurfaces
    of low degree. <i>Geometric Methods in Algebra and Number Theory</i>.  Mathematical
    Sciences Publishers. <a href="https://doi.org/10.2140/ant.2017.11.1657">https://doi.org/10.2140/ant.2017.11.1657</a>
  chicago: Browning, Timothy D, and Pankaj Vishe. “Rational Curves on Smooth Hypersurfaces
    of Low Degree.” <i>Geometric Methods in Algebra and Number Theory</i>.  Mathematical
    Sciences Publishers, 2017. <a href="https://doi.org/10.2140/ant.2017.11.1657">https://doi.org/10.2140/ant.2017.11.1657</a>.
  ieee: T. D. Browning and P. Vishe, “Rational curves on smooth hypersurfaces of low
    degree,” <i>Geometric Methods in Algebra and Number Theory</i>, vol. 11, no. 7.  Mathematical
    Sciences Publishers, pp. 1657–1675, 2017.
  ista: Browning TD, Vishe P. 2017. Rational curves on smooth hypersurfaces of low
    degree. Geometric Methods in Algebra and Number Theory. 11(7), 1657–1675.
  mla: Browning, Timothy D., and Pankaj Vishe. “Rational Curves on Smooth Hypersurfaces
    of Low Degree.” <i>Geometric Methods in Algebra and Number Theory</i>, vol. 11,
    no. 7,  Mathematical Sciences Publishers, 2017, pp. 1657–75, doi:<a href="https://doi.org/10.2140/ant.2017.11.1657">10.2140/ant.2017.11.1657</a>.
  short: T.D. Browning, P. Vishe, Geometric Methods in Algebra and Number Theory 11
    (2017) 1657–1675.
corr_author: '1'
date_created: 2018-12-11T11:45:30Z
date_published: 2017-09-07T00:00:00Z
date_updated: 2024-10-09T20:58:17Z
day: '07'
doi: 10.2140/ant.2017.11.1657
extern: '1'
external_id:
  arxiv:
  - '1611.00553'
intvolume: '        11'
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1611.00553
month: '09'
oa: 1
oa_version: Preprint
page: 1657 - 1675
publication: Geometric Methods in Algebra and Number Theory
publication_identifier:
  eissn:
  - 1944-7833
publication_status: published
publisher: ' Mathematical Sciences Publishers'
publist_id: '7637'
quality_controlled: '1'
status: public
title: Rational curves on smooth hypersurfaces of low degree
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 11
year: '2017'
...