[{"abstract":[{"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.","lang":"eng"}],"OA_type":"diamond","ec_funded":1,"file_date_updated":"2026-02-17T13:17:00Z","doi":"10.2140/ant.2025.19.883","department":[{"_id":"TiBr"}],"publication_status":"published","publication_identifier":{"eissn":["1944-7833"]},"article_processing_charge":"No","_id":"19054","OA_place":"publisher","date_created":"2025-02-18T13:33:14Z","author":[{"id":"26ca6926-5797-11ee-9232-f8b51bd19631","last_name":"Faisant","first_name":"Loïs","full_name":"Faisant, Loïs"}],"has_accepted_license":"1","ddc":["510"],"project":[{"grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program","call_identifier":"H2020","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"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.","intvolume":"        19","arxiv":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":19,"year":"2025","month":"04","language":[{"iso":"eng"}],"date_published":"2025-04-22T00:00:00Z","license":"https://creativecommons.org/licenses/by/4.0/","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"date_updated":"2026-02-17T13:19:19Z","publisher":"Mathematical Sciences Publishers","type":"journal_article","PlanS_conform":"1","publication":"Algebra & Number Theory","corr_author":"1","title":"Motivic distribution of rational curves and twisted products of toric varieties","external_id":{"arxiv":["2302.07339"]},"day":"22","oa":1,"quality_controlled":"1","oa_version":"Published Version","file":[{"file_name":"2025_AlgebraNumberTheory_Faisant.pdf","file_id":"21307","success":1,"date_updated":"2026-02-17T13:17:00Z","access_level":"open_access","relation":"main_file","creator":"dernst","date_created":"2026-02-17T13:17:00Z","file_size":2034433,"content_type":"application/pdf","checksum":"56299f55682528a7cd0136497ce8b383"}],"citation":{"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>","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>","ista":"Faisant L. 2025. Motivic distribution of rational curves and twisted products of toric varieties. Algebra &#38; Number Theory. 19, 883–965.","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>.","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.","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."},"article_type":"original","page":"883-965","status":"public"},{"file_date_updated":"2026-02-17T11:56:20Z","abstract":[{"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. ","lang":"eng"}],"OA_type":"diamond","department":[{"_id":"TiBr"}],"doi":"10.2140/ant.2025.19.2049","publication_status":"published","_id":"21244","article_processing_charge":"No","publication_identifier":{"issn":["1937-0652"],"eissn":["1944-7833"]},"OA_place":"publisher","date_created":"2026-02-16T15:22:19Z","issue":"10","author":[{"first_name":"Timothy D","last_name":"Browning","orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","full_name":"Browning, Timothy D"},{"last_name":"Lyczak","first_name":"Julian","full_name":"Lyczak, Julian"},{"full_name":"Smeets, Arne","last_name":"Smeets","first_name":"Arne"}],"has_accepted_license":"1","ddc":["510"],"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.","intvolume":"        19","project":[{"_id":"26AEDAB2-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"P32428","name":"New frontiers of the Manin conjecture"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","arxiv":1,"volume":19,"month":"09","year":"2025","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"date_updated":"2026-02-17T11:59:57Z","language":[{"iso":"eng"}],"date_published":"2025-09-05T00:00:00Z","publisher":"Mathematical Sciences Publishers","type":"journal_article","publication":"Algebra & Number Theory","PlanS_conform":"1","corr_author":"1","title":"Paucity of rational points on fibrations with multiple fibres","oa":1,"external_id":{"arxiv":["2310.01135"]},"day":"05","quality_controlled":"1","oa_version":"Published Version","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>","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.","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>.","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.","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."},"article_type":"original","file":[{"file_size":1505580,"content_type":"application/pdf","checksum":"e50a60a4303b81563f7adbcadbe2e986","file_name":"2025_AlgebraNumberTheory_Browning.pdf","date_updated":"2026-02-17T11:56:20Z","file_id":"21300","success":1,"access_level":"open_access","relation":"main_file","creator":"dernst","date_created":"2026-02-17T11:56:20Z"}],"page":"2049-2090","status":"public"},{"publisher":"Mathematical Sciences Publishers","type":"journal_article","publication":"Algebra & Number Theory","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","arxiv":1,"volume":18,"month":"02","year":"2024","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"date_updated":"2024-08-21T06:58:43Z","date_published":"2024-02-06T00:00:00Z","language":[{"iso":"eng"}],"citation":{"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.","short":"M. Pandey, V. Wang, M.W. Xu, Algebra &#38; Number Theory 18 (2024) 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>.","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.","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>.","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>"},"article_type":"original","file":[{"checksum":"1e3467a14de754bf8d3bff03a015e1ce","content_type":"application/pdf","file_size":1401725,"relation":"main_file","access_level":"open_access","date_updated":"2024-08-21T06:46:56Z","success":1,"file_name":"2024_AlgebraNumberTheory_Pandey.pdf","file_id":"17455","date_created":"2024-08-21T06:46:56Z","creator":"dernst"}],"page":"389-408","status":"public","title":"Partial sums of typical multiplicative functions over short moving intervals","oa":1,"external_id":{"arxiv":["2207.11758"]},"day":"06","oa_version":"Published Version","quality_controlled":"1","publication_status":"published","_id":"17449","article_processing_charge":"Yes (via OA deal)","publication_identifier":{"eissn":["1944-7833"],"issn":["1937-0652"]},"scopus_import":"1","extern":"1","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."}],"file_date_updated":"2024-08-21T06:46:56Z","doi":"10.2140/ant.2024.18.389","intvolume":"        18","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.","date_created":"2024-08-20T08:48:26Z","issue":"2","author":[{"last_name":"Pandey","first_name":"Mayank","full_name":"Pandey, Mayank"},{"full_name":"Wang, Victor","orcid":"0000-0002-0704-7026","last_name":"Wang","id":"76096395-aea4-11ed-a680-ab8ebbd3f1b9","first_name":"Victor"},{"first_name":"Max Wenqiang","last_name":"Xu","full_name":"Xu, Max Wenqiang"}],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2207.11758","open_access":"1"}],"has_accepted_license":"1","ddc":["510"]},{"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."}],"file_date_updated":"2023-05-30T08:05:22Z","scopus_import":"1","department":[{"_id":"TiBr"}],"doi":"10.2140/ant.2023.17.719","publication_status":"published","_id":"13091","article_processing_charge":"No","publication_identifier":{"eissn":["1944-7833"],"issn":["1937-0652"]},"author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","orcid":"0000-0002-8314-0177","first_name":"Timothy D","full_name":"Browning, Timothy D"},{"full_name":"Sawin, Will","last_name":"Sawin","first_name":"Will"}],"date_created":"2023-05-28T22:01:02Z","issue":"3","ddc":["510"],"has_accepted_license":"1","intvolume":"        17","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.","project":[{"_id":"26A8D266-B435-11E9-9278-68D0E5697425","grant_number":"EP-P026710-2","name":"Between rational and integral points"}],"volume":17,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","arxiv":1,"date_updated":"2025-04-14T09:25:44Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"date_published":"2023-04-12T00:00:00Z","language":[{"iso":"eng"}],"month":"04","year":"2023","isi":1,"publisher":"Mathematical Sciences Publishers","publication":"Algebra and Number Theory","type":"journal_article","title":"Free rational curves on low degree hypersurfaces and the circle method","corr_author":"1","oa_version":"Published Version","quality_controlled":"1","oa":1,"day":"12","external_id":{"isi":["000996014700004"],"arxiv":["1810.06882"]},"page":"719-748","article_type":"original","citation":{"short":"T.D. Browning, W. Sawin, Algebra and Number Theory 17 (2023) 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>.","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.","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>","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>","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>.","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."},"file":[{"file_size":1430719,"content_type":"application/pdf","checksum":"5d5d67b235905650e33cf7065d7583b4","creator":"dernst","date_created":"2023-05-30T08:05:22Z","file_id":"13101","file_name":"2023_AlgebraNumberTheory_Browning.pdf","access_level":"open_access","success":1,"date_updated":"2023-05-30T08:05:22Z","relation":"main_file"}],"status":"public"},{"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2102.11552"}],"issue":"10","date_created":"2021-02-25T09:56:57Z","author":[{"full_name":"Browning, Timothy D","first_name":"Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","orcid":"0000-0002-8314-0177"},{"full_name":"Horesh, Tal","id":"C8B7BF48-8D81-11E9-BCA9-F536E6697425","last_name":"Horesh","first_name":"Tal"},{"full_name":"Wilsch, Florian Alexander","first_name":"Florian Alexander","orcid":"0000-0001-7302-8256","last_name":"Wilsch","id":"560601DA-8D36-11E9-A136-7AC1E5697425"}],"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","call_identifier":"FWF","_id":"26AEDAB2-B435-11E9-9278-68D0E5697425"}],"intvolume":"        16","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.","doi":"10.2140/ant.2022.16.2385","department":[{"_id":"TiBr"}],"scopus_import":"1","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"}],"publication_identifier":{"issn":["1937-0652"],"eissn":["1944-7833"]},"article_processing_charge":"No","_id":"9199","publication_status":"published","external_id":{"isi":["000961514100004"],"arxiv":["2102.11552"]},"day":"01","oa":1,"quality_controlled":"1","oa_version":"Preprint","corr_author":"1","title":"Equidistribution and freeness on Grassmannians","status":"public","article_type":"original","citation":{"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.","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>.","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>.","ista":"Browning TD, Horesh T, Wilsch FA. 2022. Equidistribution and freeness on Grassmannians. Algebra &#38; Number Theory. 16(10), 2385–2407.","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>","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>"},"page":"2385-2407","year":"2022","month":"12","date_published":"2022-12-01T00:00:00Z","language":[{"iso":"eng"}],"date_updated":"2025-04-14T09:25:44Z","arxiv":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","volume":16,"type":"journal_article","publication":"Algebra & Number Theory","publisher":"Mathematical Sciences Publishers","isi":1},{"intvolume":"        15","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1910.05207"}],"author":[{"id":"98C47862-10D5-11EA-BEDD-0F6F3DDC885E","last_name":"Bilu","first_name":"Margaret","full_name":"Bilu, Margaret"},{"full_name":"Howe, Sean","first_name":"Sean","last_name":"Howe"}],"date_created":"2024-04-03T08:12:59Z","issue":"9","article_processing_charge":"No","_id":"15279","publication_identifier":{"issn":["1937-0652"],"eissn":["1944-7833"]},"keyword":["Algebra and Number Theory"],"publication_status":"published","department":[{"_id":"TiBr"}],"doi":"10.2140/ant.2021.15.2195","abstract":[{"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.","lang":"eng"}],"scopus_import":"1","status":"public","page":"2195-2259","citation":{"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.","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.","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>.","ista":"Bilu M, Howe S. 2021. Motivic Euler products in motivic statistics. Algebra &#38; Number Theory. 15(9), 2195–2259.","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>","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>"},"article_type":"original","oa_version":"Preprint","quality_controlled":"1","oa":1,"day":"23","external_id":{"arxiv":["1910.05207"]},"title":"Motivic Euler products in motivic statistics","corr_author":"1","publication":"Algebra & Number Theory","type":"journal_article","publisher":"Mathematical Sciences Publishers","date_updated":"2024-10-21T06:02:15Z","language":[{"iso":"eng"}],"date_published":"2021-12-23T00:00:00Z","month":"12","year":"2021","volume":15,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","arxiv":1},{"publisher":" Mathematical Sciences Publishers","type":"journal_article","publication":"Geometric Methods in Algebra and Number Theory","arxiv":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":11,"year":"2017","month":"09","language":[{"iso":"eng"}],"date_published":"2017-09-07T00:00:00Z","date_updated":"2024-10-09T20:58:17Z","citation":{"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.","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.","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>","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>","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>.","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."},"article_type":"original","page":"1657 - 1675","status":"public","corr_author":"1","title":"Rational curves on smooth hypersurfaces of low degree","day":"07","external_id":{"arxiv":["1611.00553"]},"oa":1,"quality_controlled":"1","oa_version":"Preprint","publist_id":"7637","publication_status":"published","publication_identifier":{"eissn":["1944-7833"]},"article_processing_charge":"No","_id":"265","extern":"1","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."}],"doi":"10.2140/ant.2017.11.1657","intvolume":"        11","acknowledgement":"While working on this paper the first author was supported by ERC grant 306457.","issue":"7","date_created":"2018-12-11T11:45:30Z","author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","orcid":"0000-0002-8314-0177","first_name":"Timothy D","full_name":"Browning, Timothy D"},{"full_name":"Vishe, Pankaj","last_name":"Vishe","first_name":"Pankaj"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1611.00553"}]}]