[{"oa_version":"Published Version","has_accepted_license":"1","publication_status":"published","OA_type":"hybrid","researchdata_availability":"yes","type":"journal_article","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).","department":[{"_id":"TiBr"}],"scopus_import":"1","file_date_updated":"2025-09-03T06:44:44Z","corr_author":"1","citation":{"short":"T.D. Browning, F.A. Wilsch, Selecta Mathematica New Series 31 (2025).","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>.","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>.","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>","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>","ista":"Browning TD, Wilsch FA. 2025. Integral points on cubic surfaces: heuristics and numerics. Selecta Mathematica New Series. 31(4), 81.","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."},"volume":31,"related_material":{"link":[{"relation":"software","url":"https://github.com/fwilsch/cubicpts"}],"record":[{"relation":"research_data","id":"22234","status":"public"}]},"year":"2025","OA_place":"publisher","doi":"10.1007/s00029-025-01074-1","issue":"4","isi":1,"author":[{"full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D","last_name":"Browning"},{"full_name":"Wilsch, Florian Alexander","orcid":"0000-0001-7302-8256","last_name":"Wilsch","id":"560601DA-8D36-11E9-A136-7AC1E5697425","first_name":"Florian Alexander"}],"external_id":{"arxiv":["2407.16315"],"isi":["001552779800001"]},"das_tickbox":"1","publication":"Selecta Mathematica New Series","publication_identifier":{"eissn":["1420-9020"],"issn":["1022-1824"]},"project":[{"_id":"26AEDAB2-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"New frontiers of the Manin conjecture","grant_number":"P32428"},{"_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","name":"Rational curves via function field analytic number theory","grant_number":"P36278"}],"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."}],"file":[{"file_name":"2025_SelectaMathematica_Browning.pdf","date_created":"2025-09-03T06:44:44Z","date_updated":"2025-09-03T06:44:44Z","relation":"main_file","access_level":"open_access","file_id":"20281","checksum":"89352f1f7e8d2b367ae5f4e9bf9eb1f5","content_type":"application/pdf","creator":"dernst","success":1,"file_size":2484757}],"language":[{"iso":"eng"}],"date_published":"2025-09-01T00:00:00Z","article_type":"original","article_number":"81","quality_controlled":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"status":"public","day":"01","arxiv":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.","title":"Integral points on cubic surfaces: heuristics and numerics","intvolume":"        31","oa":1,"date_created":"2025-08-31T22:01:31Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2026-08-12T12:15:32Z","supplementarymaterial":"yes","article_processing_charge":"Yes (via OA deal)","_id":"20249","ddc":["500"],"PlanS_conform":"1","fulldoi":"https://doi.org/10.1007/s00029-025-01074-1","month":"09","publisher":"Springer Nature"},{"supplementarymaterial":"no","keyword":["Integral points","del Pezzo surface","universal torsor","Manin’s conjecture"],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2021-09-15T10:06:48Z","date_updated":"2026-07-29T10:00:51Z","oa":1,"ddc":["510"],"publisher":"Cambridge University Press","fulldoi":"https://doi.org/10.1017/S1474748022000482","month":"05","article_processing_charge":"Yes (via OA deal)","_id":"10018","status":"public","article_type":"original","quality_controlled":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"arxiv":1,"intvolume":"        23","title":"Integral points on singular del Pezzo surfaces","day":"10","isi":1,"issue":"3","page":"1259-1294","author":[{"full_name":"Derenthal, Ulrich","last_name":"Derenthal","first_name":"Ulrich"},{"orcid":"0000-0001-7302-8256","first_name":"Florian Alexander","last_name":"Wilsch","id":"560601DA-8D36-11E9-A136-7AC1E5697425","full_name":"Wilsch, Florian Alexander"}],"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."}],"language":[{"iso":"eng"}],"date_published":"2024-05-10T00:00:00Z","file":[{"success":1,"creator":"dernst","content_type":"application/pdf","file_size":592305,"file_name":"2024_JourMathJussieu_Derenthal.pdf","date_created":"2024-06-03T08:33:29Z","date_updated":"2024-06-03T08:33:29Z","access_level":"open_access","file_id":"17102","relation":"main_file","checksum":"c4698ea12cfe10ef2c6c79880290c7ac"}],"publication":"Journal of the Institute of Mathematics of Jussieu","external_id":{"isi":["000881319200001"],"arxiv":["2109.06778"]},"das_tickbox":"0","project":[{"call_identifier":"FWF","_id":"26AEDAB2-B435-11E9-9278-68D0E5697425","grant_number":"P32428","name":"New frontiers of the Manin conjecture"}],"publication_identifier":{"eissn":["1475-3030 "],"issn":["1474-7480"]},"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.","researchdata_availability":"no","type":"journal_article","scopus_import":"1","department":[{"_id":"TiBr"}],"oa_version":"Published Version","publication_status":"published","has_accepted_license":"1","doi":"10.1017/S1474748022000482","year":"2024","corr_author":"1","file_date_updated":"2024-06-03T08:33:29Z","volume":23,"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>","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.","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.","short":"U. Derenthal, F.A. Wilsch, Journal of the Institute of Mathematics of Jussieu 23 (2024) 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>.","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>."}},{"abstract":[{"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.","lang":"eng"}],"language":[{"iso":"eng"}],"date_published":"2023-04-01T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1901.08503"}],"publication":"International Mathematics Research Notices","das_tickbox":"0","external_id":{"arxiv":["1901.08503"],"isi":["000773116000001"]},"publication_identifier":{"eissn":["1687-0247"],"issn":["1073-7928"]},"isi":1,"page":"6780-6808","issue":"8","author":[{"full_name":"Wilsch, Florian Alexander","orcid":"0000-0001-7302-8256","last_name":"Wilsch","id":"560601DA-8D36-11E9-A136-7AC1E5697425","first_name":"Florian Alexander"}],"doi":"10.1093/imrn/rnac048","year":"2023","corr_author":"1","volume":2023,"citation":{"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>.","short":"F.A. Wilsch, International Mathematics Research Notices 2023 (2023) 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>.","ista":"Wilsch FA. 2023. Integral points of bounded height on a log Fano threefold. International Mathematics Research Notices. 2023(8), 6780–6808.","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.","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>"},"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.","researchdata_availability":"no","type":"journal_article","scopus_import":"1","department":[{"_id":"TiBr"}],"oa_version":"Preprint","publication_status":"published","publisher":"Oxford University Press","fulldoi":"https://doi.org/10.1093/imrn/rnac048","month":"04","article_processing_charge":"No","_id":"9034","supplementarymaterial":"no","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_updated":"2026-07-29T10:43:18Z","date_created":"2021-01-22T09:31:09Z","oa":1,"arxiv":1,"intvolume":"      2023","title":"Integral points of bounded height on a log Fano threefold","day":"01","status":"public","article_type":"original","quality_controlled":"1"},{"year":"2022","doi":"10.48550/arXiv.2202.10909","citation":{"ista":"Wilsch FA. Integral points of bounded height on a certain toric variety. arXiv, 2202.10909.","ieee":"F. A. Wilsch, “Integral points of bounded height on a certain toric variety,” <i>arXiv</i>. .","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>.","short":"F.A. Wilsch, ArXiv (n.d.).","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>."},"corr_author":"1","department":[{"_id":"TiBr"}],"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.","researchdata_availability":"no","type":"preprint","publication_status":"submitted","oa_version":"Preprint","date_published":"2022-02-22T00:00:00Z","language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2202.10909"}],"abstract":[{"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.","lang":"eng"}],"project":[{"name":"New frontiers of the Manin conjecture","grant_number":"P32428","call_identifier":"FWF","_id":"26AEDAB2-B435-11E9-9278-68D0E5697425"}],"publication":"arXiv","das_tickbox":"0","external_id":{"arxiv":["2202.10909"]},"author":[{"orcid":"0000-0001-7302-8256","id":"560601DA-8D36-11E9-A136-7AC1E5697425","last_name":"Wilsch","first_name":"Florian Alexander","full_name":"Wilsch, Florian Alexander"}],"title":"Integral points of bounded height on a certain toric variety","arxiv":1,"day":"22","status":"public","article_number":"2202.10909","fulldoi":"https://doi.org/10.48550/arXiv.2202.10909","month":"02","_id":"10788","article_processing_charge":"No","supplementarymaterial":"no","keyword":["Integral point","toric variety","Manin's conjecture"],"date_updated":"2026-08-06T10:31:31Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2022-02-23T09:04:43Z","oa":1},{"isi":1,"issue":"10","page":"2385-2407","author":[{"first_name":"Timothy D","last_name":"Browning","id":"35827D50-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8314-0177","full_name":"Browning, Timothy D"},{"id":"C8B7BF48-8D81-11E9-BCA9-F536E6697425","last_name":"Horesh","first_name":"Tal","full_name":"Horesh, Tal"},{"full_name":"Wilsch, Florian Alexander","first_name":"Florian Alexander","last_name":"Wilsch","id":"560601DA-8D36-11E9-A136-7AC1E5697425","orcid":"0000-0001-7302-8256"}],"publication":"Algebra & Number Theory","external_id":{"isi":["000961514100004"],"arxiv":["2102.11552"]},"das_tickbox":"0","publication_identifier":{"issn":["1937-0652"],"eissn":["1944-7833"]},"project":[{"name":"Between rational and integral points","grant_number":"EP-P026710-2","_id":"26A8D266-B435-11E9-9278-68D0E5697425"},{"name":"New frontiers of the Manin conjecture","grant_number":"P32428","_id":"26AEDAB2-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"abstract":[{"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.","lang":"eng"}],"language":[{"iso":"eng"}],"date_published":"2022-12-01T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2102.11552"}],"oa_version":"Preprint","publication_status":"published","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.","researchdata_availability":"no","type":"journal_article","scopus_import":"1","department":[{"_id":"TiBr"}],"corr_author":"1","volume":16,"citation":{"ista":"Browning TD, Horesh T, Wilsch FA. 2022. Equidistribution and freeness on Grassmannians. Algebra &#38; Number Theory. 16(10), 2385–2407.","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.","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>","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>.","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>","short":"T.D. Browning, T. Horesh, F.A. Wilsch, Algebra &#38; Number Theory 16 (2022) 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>."},"year":"2022","doi":"10.2140/ant.2022.16.2385","oa":1,"supplementarymaterial":"no","date_created":"2021-02-25T09:56:57Z","date_updated":"2026-08-06T11:07:27Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","article_processing_charge":"No","_id":"9199","publisher":"Mathematical Sciences Publishers","fulldoi":"https://doi.org/10.2140/ant.2022.16.2385","month":"12","article_type":"original","quality_controlled":"1","status":"public","day":"01","arxiv":1,"intvolume":"        16","title":"Equidistribution and freeness on Grassmannians"}]