---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20249'
abstract:
- lang: eng
  text: We develop a heuristic for the density of integer points on affine cubic surfaces.
    Our heuristic applies to smooth surfaces defined by cubic polynomials that are
    log K3, but it can also be adjusted to handle singular cubic surfaces. We compare
    our heuristic to Heath-Brown’s prediction for sums of three cubes, as well as
    to asymptotic formulae in the literature around Zagier’s work on the Markoff cubic
    surface, and work of Baragar and Umeda on further surfaces of Markoff-type. We
    also test our heuristic against numerical data for several families of cubic surfaces.
acknowledgement: "The authors owe a debt of thanks to Yonatan Harpaz for asking about
  circle method heuristics for log K3 surfaces. His contribution to the resulting
  discussion is gratefully acknowledged. Thanks are also due to Andrew Sutherland
  for help with numerical data for the equation x^3 + y^3 + z^3 = 1, together with
  Alex Gamburd, Amit Ghosh, Peter Sarnak and Matteo Verzobio for their interest in
  this paper. Special thanks are due to Victor Wang for helpful conversations about
  the circle method heuristics and to the anonymous referee for several useful comments.
  While working on this paper, the authors were supported by a FWF grant (DOI 10.55776/P32428),
  and the first author was supported by a further FWF grant (DOI 10.55776/P36278)
  and a grant from the School of Mathematics at the Institute for Advanced Study in
  Princeton.\r\nOpen access funding provided by Institute of Science and Technology
  (IST Austria)."
article_number: '81'
article_processing_charge: Yes (via OA deal)
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: 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, Wilsch FA. Integral points on cubic surfaces: heuristics and
    numerics. <i>Selecta Mathematica New Series</i>. 2025;31(4). doi:<a href="https://doi.org/10.1007/s00029-025-01074-1">10.1007/s00029-025-01074-1</a>'
  apa: 'Browning, T. D., &#38; Wilsch, F. A. (2025). Integral points on cubic surfaces:
    heuristics and numerics. <i>Selecta Mathematica New Series</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s00029-025-01074-1">https://doi.org/10.1007/s00029-025-01074-1</a>'
  chicago: 'Browning, Timothy D, and Florian Alexander Wilsch. “Integral Points on
    Cubic Surfaces: Heuristics and Numerics.” <i>Selecta Mathematica New Series</i>.
    Springer Nature, 2025. <a href="https://doi.org/10.1007/s00029-025-01074-1">https://doi.org/10.1007/s00029-025-01074-1</a>.'
  ieee: 'T. D. Browning and F. A. Wilsch, “Integral points on cubic surfaces: heuristics
    and numerics,” <i>Selecta Mathematica New Series</i>, vol. 31, no. 4. Springer
    Nature, 2025.'
  ista: 'Browning TD, Wilsch FA. 2025. Integral points on cubic surfaces: heuristics
    and numerics. Selecta Mathematica New Series. 31(4), 81.'
  mla: 'Browning, Timothy D., and Florian Alexander Wilsch. “Integral Points on Cubic
    Surfaces: Heuristics and Numerics.” <i>Selecta Mathematica New Series</i>, vol.
    31, no. 4, 81, Springer Nature, 2025, doi:<a href="https://doi.org/10.1007/s00029-025-01074-1">10.1007/s00029-025-01074-1</a>.'
  short: T.D. Browning, F.A. Wilsch, Selecta Mathematica New Series 31 (2025).
corr_author: '1'
das_tickbox: '1'
dataavailabilitystatement: The data used in Sections 6 and 9 is hosted on the Göttingen
  Research Online Data repository [https://doi.org/10.25625/4FLFH8]. The code used
  to determine the data in Sections 6.2, 9.1 and 9.2 is found on the second author’s
  github page.
date_created: 2025-08-31T22:01:31Z
date_published: 2025-09-01T00:00:00Z
date_updated: 2026-07-17T11:59:39Z
day: '01'
ddc:
- '500'
department:
- _id: TiBr
doi: 10.1007/s00029-025-01074-1
external_id:
  arxiv:
  - '2407.16315'
  isi:
  - '001552779800001'
file:
- access_level: open_access
  checksum: 89352f1f7e8d2b367ae5f4e9bf9eb1f5
  content_type: application/pdf
  creator: dernst
  date_created: 2025-09-03T06:44:44Z
  date_updated: 2025-09-03T06:44:44Z
  file_id: '20281'
  file_name: 2025_SelectaMathematica_Browning.pdf
  file_size: 2484757
  relation: main_file
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file_date_updated: 2025-09-03T06:44:44Z
has_accepted_license: '1'
intvolume: '        31'
isi: 1
issue: '4'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
project:
- _id: 26AEDAB2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P32428
  name: New frontiers of the Manin conjecture
- _id: bd8a4fdc-d553-11ed-ba76-80a0167441a3
  grant_number: P36278
  name: Rational curves via function field analytic number theory
publication: Selecta Mathematica New Series
publication_identifier:
  eissn:
  - 1420-9020
  issn:
  - 1022-1824
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
  record:
  - id: '22234'
    relation: research_data
    status: for_moderation
  - id: '22235'
    relation: research_data
    status: for_moderation
researchdata_availability: yes
scopus_import: '1'
status: public
supplementarymaterial: yes
title: 'Integral points on cubic surfaces: heuristics and numerics'
tmp:
  image: /images/cc_by.png
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  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 31
year: '2025'
...
---
_id: '10018'
abstract:
- lang: eng
  text: In order to study integral points of bounded log-anticanonical height on weak
    del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example,
    we consider a quartic del Pezzo surface of singularity type A1 + A3 and prove
    an analogue of Manin's conjecture for integral points with respect to its singularities
    and its lines.
acknowledgement: The first author was partly supported by grant DE 1646/4-2 of the
  Deutsche Forschungsgemeinschaft. The second author was partly supported by FWF grant
  P 32428-N35 and conducted part of this work as a guest at the Institut de Mathématiques
  de Jussieu–Paris Rive Gauche invited by Antoine Chambert-Loir and funded by DAAD.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Ulrich
  full_name: Derenthal, Ulrich
  last_name: Derenthal
- 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: Derenthal U, Wilsch FA. Integral points on singular del Pezzo surfaces. <i>Journal
    of the Institute of Mathematics of Jussieu</i>. 2024;23(3):1259-1294. doi:<a href="https://doi.org/10.1017/S1474748022000482">10.1017/S1474748022000482</a>
  apa: Derenthal, U., &#38; Wilsch, F. A. (2024). Integral points on singular del
    Pezzo surfaces. <i>Journal of the Institute of Mathematics of Jussieu</i>. Cambridge
    University Press. <a href="https://doi.org/10.1017/S1474748022000482">https://doi.org/10.1017/S1474748022000482</a>
  chicago: Derenthal, Ulrich, and Florian Alexander Wilsch. “Integral Points on Singular
    Del Pezzo Surfaces.” <i>Journal of the Institute of Mathematics of Jussieu</i>.
    Cambridge University Press, 2024. <a href="https://doi.org/10.1017/S1474748022000482">https://doi.org/10.1017/S1474748022000482</a>.
  ieee: U. Derenthal and F. A. Wilsch, “Integral points on singular del Pezzo surfaces,”
    <i>Journal of the Institute of Mathematics of Jussieu</i>, vol. 23, no. 3. Cambridge
    University Press, pp. 1259–1294, 2024.
  ista: Derenthal U, Wilsch FA. 2024. Integral points on singular del Pezzo surfaces.
    Journal of the Institute of Mathematics of Jussieu. 23(3), 1259–1294.
  mla: Derenthal, Ulrich, and Florian Alexander Wilsch. “Integral Points on Singular
    Del Pezzo Surfaces.” <i>Journal of the Institute of Mathematics of Jussieu</i>,
    vol. 23, no. 3, Cambridge University Press, 2024, pp. 1259–94, doi:<a href="https://doi.org/10.1017/S1474748022000482">10.1017/S1474748022000482</a>.
  short: U. Derenthal, F.A. Wilsch, Journal of the Institute of Mathematics of Jussieu
    23 (2024) 1259–1294.
corr_author: '1'
date_created: 2021-09-15T10:06:48Z
date_published: 2024-05-10T00:00:00Z
date_updated: 2025-04-15T07:39:01Z
day: '10'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1017/S1474748022000482
external_id:
  arxiv:
  - '2109.06778'
  isi:
  - '000881319200001'
file:
- access_level: open_access
  checksum: c4698ea12cfe10ef2c6c79880290c7ac
  content_type: application/pdf
  creator: dernst
  date_created: 2024-06-03T08:33:29Z
  date_updated: 2024-06-03T08:33:29Z
  file_id: '17102'
  file_name: 2024_JourMathJussieu_Derenthal.pdf
  file_size: 592305
  relation: main_file
  success: 1
file_date_updated: 2024-06-03T08:33:29Z
has_accepted_license: '1'
intvolume: '        23'
isi: 1
issue: '3'
keyword:
- Integral points
- del Pezzo surface
- universal torsor
- Manin’s conjecture
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: 1259-1294
project:
- _id: 26AEDAB2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P32428
  name: New frontiers of the Manin conjecture
publication: Journal of the Institute of Mathematics of Jussieu
publication_identifier:
  eissn:
  - '1475-3030 '
  issn:
  - 1474-7480
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Integral points on singular del Pezzo surfaces
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 23
year: '2024'
...
---
_id: '9034'
abstract:
- lang: eng
  text: We determine an asymptotic formula for the number of integral points of bounded
    height on a blow-up of P3 outside certain planes using universal torsors.
acknowledgement: This work was supported by the German Academic Exchange Service.
  Parts of this article were prepared at the Institut de Mathémathiques de Jussieu—Paris
  Rive Gauche. I wish to thank Antoine Chambert-Loir for his remarks and the institute
  for its hospitality, as well as the anonymous referee for several useful remarks
  and suggestions for improvements.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- 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: Wilsch FA. Integral points of bounded height on a log Fano threefold. <i>International
    Mathematics Research Notices</i>. 2023;2023(8):6780-6808. doi:<a href="https://doi.org/10.1093/imrn/rnac048">10.1093/imrn/rnac048</a>
  apa: Wilsch, F. A. (2023). Integral points of bounded height on a log Fano threefold.
    <i>International Mathematics Research Notices</i>. Oxford University Press. <a
    href="https://doi.org/10.1093/imrn/rnac048">https://doi.org/10.1093/imrn/rnac048</a>
  chicago: Wilsch, Florian Alexander. “Integral Points of Bounded Height on a Log
    Fano Threefold.” <i>International Mathematics Research Notices</i>. Oxford University
    Press, 2023. <a href="https://doi.org/10.1093/imrn/rnac048">https://doi.org/10.1093/imrn/rnac048</a>.
  ieee: F. A. Wilsch, “Integral points of bounded height on a log Fano threefold,”
    <i>International Mathematics Research Notices</i>, vol. 2023, no. 8. Oxford University
    Press, pp. 6780–6808, 2023.
  ista: Wilsch FA. 2023. Integral points of bounded height on a log Fano threefold.
    International Mathematics Research Notices. 2023(8), 6780–6808.
  mla: Wilsch, Florian Alexander. “Integral Points of Bounded Height on a Log Fano
    Threefold.” <i>International Mathematics Research Notices</i>, vol. 2023, no.
    8, Oxford University Press, 2023, pp. 6780–808, doi:<a href="https://doi.org/10.1093/imrn/rnac048">10.1093/imrn/rnac048</a>.
  short: F.A. Wilsch, International Mathematics Research Notices 2023 (2023) 6780–6808.
corr_author: '1'
date_created: 2021-01-22T09:31:09Z
date_published: 2023-04-01T00:00:00Z
date_updated: 2025-05-14T11:07:41Z
day: '01'
department:
- _id: TiBr
doi: 10.1093/imrn/rnac048
external_id:
  arxiv:
  - '1901.08503'
  isi:
  - '000773116000001'
intvolume: '      2023'
isi: 1
issue: '8'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1901.08503
month: '04'
oa: 1
oa_version: Preprint
page: 6780-6808
publication: International Mathematics Research Notices
publication_identifier:
  eissn:
  - 1687-0247
  issn:
  - 1073-7928
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Integral points of bounded height on a log Fano threefold
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2023
year: '2023'
...
---
_id: '10788'
abstract:
- lang: eng
  text: "We determine an asymptotic formula for the number of integral points of\r\nbounded
    height on a certain toric variety, which is incompatible with part of a\r\npreprint
    by Chambert-Loir and Tschinkel. We provide an alternative\r\ninterpretation of
    the asymptotic formula we get. To do so, we construct an\r\nanalogue of Peyre's
    constant $\\alpha$ and describe its relation to a new\r\nobstruction to the Zariski
    density of integral points in certain regions of\r\nvarieties."
acknowledgement: "Part of this work was conducted as a guest at the Institut de Mathématiques
  de Jussieu–Paris Rive Gauche invited by Antoine Chambert-Loir and funded by DAAD.\r\nDuring
  this time, I had interesting and fruitful discussions on the interpretation of the
  result for\r\nthe toric variety discussed in Section 3 with Antoine Chambert-Loir.
  I wish to thank him for these\r\nopportunities and for his useful remarks on earlier
  versions of this article. This work was partly\r\nfunded by FWF grant P 32428-N35."
article_number: '2202.10909'
article_processing_charge: No
arxiv: 1
author:
- 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: Wilsch FA. Integral points of bounded height on a certain toric variety. <i>arXiv</i>.
    doi:<a href="https://doi.org/10.48550/arXiv.2202.10909">10.48550/arXiv.2202.10909</a>
  apa: Wilsch, F. A. (n.d.). Integral points of bounded height on a certain toric
    variety. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2202.10909">https://doi.org/10.48550/arXiv.2202.10909</a>
  chicago: Wilsch, Florian Alexander. “Integral Points of Bounded Height on a Certain
    Toric Variety.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2202.10909">https://doi.org/10.48550/arXiv.2202.10909</a>.
  ieee: F. A. Wilsch, “Integral points of bounded height on a certain toric variety,”
    <i>arXiv</i>. .
  ista: Wilsch FA. Integral points of bounded height on a certain toric variety. arXiv,
    2202.10909.
  mla: Wilsch, Florian Alexander. “Integral Points of Bounded Height on a Certain
    Toric Variety.” <i>ArXiv</i>, 2202.10909, doi:<a href="https://doi.org/10.48550/arXiv.2202.10909">10.48550/arXiv.2202.10909</a>.
  short: F.A. Wilsch, ArXiv (n.d.).
corr_author: '1'
date_created: 2022-02-23T09:04:43Z
date_published: 2022-02-22T00:00:00Z
date_updated: 2025-04-15T07:39:01Z
day: '22'
department:
- _id: TiBr
doi: 10.48550/arXiv.2202.10909
external_id:
  arxiv:
  - '2202.10909'
keyword:
- Integral point
- toric variety
- Manin's conjecture
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2202.10909
month: '02'
oa: 1
oa_version: Preprint
project:
- _id: 26AEDAB2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P32428
  name: New frontiers of the Manin conjecture
publication: arXiv
publication_status: submitted
status: public
title: Integral points of bounded height on a certain toric variety
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2022'
...
---
_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'
...